Isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature

We prove a sharp isoperimetric inequality for measured Finsler manifolds having non-negative Ricci curvature and Euclidean volume growth. We also prove a rigidity result for this inequality, under the additional hypotheses of boundedness of the isoperimetric set and the finite reversibility of the space. As a consequence, we deduce the rigidity of the weighted anisotropic isoperimetric inequality for cones in the Euclidean space, in the irreversible setting.


Introduction
The classical isoperimetric inequality in the Eucludean space states that if E is a (sufficiently regular) subset of R d , then (1.1) P(E) ≥ dω where P(E) denotes the perimeter of the set E, ω d the measure of the unit ball in R d , and L d the Lebesgue measure.Moreover, if the equality is attained in (1.1) by a certain set E with positive measure, then the set E coincides with a ball of radius ( L d (E) ω d ) 1/d .This inequality has been successfully extended in more general settings where the space is not the Euclidean one.Indeed, it turns out that two are the relevant properties of the Euclidean space needed for such generalizations: 1) the fact that R d has non-negative Ricci curvature; 2) its Euclidean volume growth, i.e. a constraint on the growth of the measure of large balls.
If (X, g) is an n-dimensional Riemannian manifold, one can consider different measures to the canonical volume Vol g .In this case, the Ricci tensor has to be replaced with the generalised N -Ricci tensor: if h : X → (0, ∞) is a weight for the volume Vol g , the generalised N -Ricci tensor, with N > n, is defined by We say that the weighted manifold (X, g, h Vol g ) verifies the so-called Curvature-Dimension condition CD(K, N ) (see [6]) whenever Ric N ≥ Kg.
In their seminal works Lott-Villani [30] and Sturm [45,44] introduced a synthetic definition of CD(K, N ) for complete and separable metric spaces (X, d) endowed with a (locally-finite Borel) reference measure m ("metric-measure space", or m.m.s.).This CD(K, N ) condition is formulated using the theory Optimal Transport and it coincides with the Bakry-Émery one in the smooth Riemannian setting (and in particular in the classical non-weighted one).
A measured Finsler manifold is a triple (X, F, m), such that X is a differential manifold (possibly with boundary), m a Borel measure, and F a Finsler structure, that is a real-valued function F : T X → [0, ∞), which is convex, positively homogeneous, and F (v) = 0 if and only if v = 0 (see Section 2.1 for the precise definition).In general F (v) = F (−v).This feature, know as irreversibility, is what prevents to apply the techniques developed for m.m.s.'s to measured Finsler manifolds.Recently, Ohta successfully extended the theory of the Curvature-Dimension condition for possibly-irreversible Finsler manifolds (see [35,38,40]).Namely, a notion of N -Ricci curvature (compatible with the riemannian one) was introduced and it was proven that a measured Finsler manifold satisfies the CD(K, N ) condition if and only if Ric N ≥ K.More recently, the notion of irreversible metric measure space has been introduced [29].
The isoperimetric problem in the reversible setting has been extensively investigated.E. Milman [32] gave a sharp isoperimetric inequality for weighted riemannian manifolds satisfying the CD(K, N ) condition (for any K ∈ R, N > 1), with an additional constraint on the diameter.In particular, given K ∈ R, N > 1, D ∈ (0, ∞], he gave an explicit description of the so-called isoperimetric profile function I K,N,D : [0, 1] → R. The isoperimetric profile has the property that, given a weighted riemannian manifold satisfying the CD(K, N ) condition with diameter at most D, whose total measure is 1, it holds that P(E) ≥ I K,N,D (v), for any subset E of measure v ∈ [0, 1]; moreover, Milman's result is sharp.Cavalletti and Mondino [14] extended Milman's result to the non-smooth setting finding the same lower bound.Their proof makes use of the localisation method (also known as needle decomposition), a powerful dimensional reduction tool, initially developed by Klartag [27] for riemannian manifolds and later extended to CD(K, N ) spaces [14].
In the setting of measured Finsler manifold, less is known.Following the line traced in [14], Ohta [39] extended the localization method to measured Finsler manifolds, obtaining a lower bound for the perimeter for measured Finsler manifolds with finite reversibility constant.The reversibility constant (introduced in [43]) of a Finsler structure F on the manifold X is defined as the least constant (possibly infinite) Λ F ≥ 1 such that F (−v) ≤ Λ F F (v) for all vectors v ∈ T X. Ohta proved [39] that given a measured Finsler manifold (X, F, m) having finite reversibility constant and m(X) = 1 satisfying the CD(K, N ) condition, with diameter bounded from above by D, it holds that (1.3) P(E) ≥ Λ −1 F I K,N,D (m(E)), ∀E ⊂ X, where I K,N,D is the isoperimetric profile function described by E. Milman.The presence of the factor Λ −1 F suggests that the inequality above is not sharp.Indeed, in the case N = D = ∞, this factor can be eliminated obtaining the Barky-Ledoux isoperimetric inequality for Finsler manifolds [41].
Regarding the case K = 0, in order to generalize the classical inequality (1.1) we must drop the assumption that the space has measure 1 and consider the case when the measure is infinite.However, it is well known that without an additional condition on the geometry of the space no non-trivial isoperimetric inequality holds in the case K = 0.A way to create an Euclidean-like environment is to impose an additional constraint on the growth of the measure of the balls that mimics the Euclidean one.Letting B + (x, r) = {y : d(x, y) < r} denoting the forward metric ball with center x ∈ X and radius r > 0, by the Bishop-Gromov inequality (see (2.23)) the map r → m(B + (r,x)) r N is nonincreasing over (0, ∞) for any x ∈ X.The asymptotic volume ratio is then naturally defined by (1.4) It is easy to see that it is indeed independent of the choice of x ∈ X.When AVR X > 0, we say that space has Euclidean volume growth.
In the riemannian setting, the isoperimetric inequality for spaces with Euclidean volume growth has been obtained in increasing generality (see, e.g.[1,23,10,25]).The most general result is due to Balogh and Kristály [7] and it is valid for (non-smooth) CD(0, N ) spaces; it was proven exploiting the Brunn-Minkowski inequality for CD(0, N ) spaces.
Theorem 1.1 ([7, Theorem 1.1]).Let (X, d, m) be a m.m.s.satisfying the CD(0, N ) condition for some N > 1 and having Euclidean volume growth.Then for every bounded Borel subset E ⊂ X it holds Moreover, inequality (1.5) is sharp.
In inequality (1.5), m + denotes the Minkowski content.In Appendix B we will discuss the relation between the Minkowski content and the perimeter; for mildly regular sets these two notions coincide.Using the Γ-function, one can naturally define the constant ω N for all N > 1.
The rigidity of the inequality has been obtained, under two mild assumptions, by the author and Cavalletti [13].These assumption are: 1) the essentially-non-branching-ness of the space, that excludes pathological cases; 2) the fact that the set attaining equality in (1.5) is bounded.
Then there exists (a unique) o ∈ X such that, up to a negligible set, E = B + (o, ρ), with ρ = ( m(E) AVR X ω N ) 1 N .Moreover, the measure m can be represented by the disintegration formula m α q(dα), q ∈ P(∂B where m α is concentrated on the geodesic ray from o through α and m α can be identified (via the unitary speed parametrisation of the ray) with N ω N AVR X t N −1 L 1 [0,∞) .
As a consequence of this result, having in mind the fact that "volume cone implies metric cone" [19], we obtain that when the space is RCD(0, N ), we also have a rigidity of the metric, i.e., the space is isometric to a cone over an RCD(N − 1, N − 2) space.In the case of non-collapsed RCD spaces, the hypothesis on the boundedness of the set can be lifted [5,4] (see also [9]).
The scope of the present paper is to extend Theorems 1.1 and 1.2 to the setting of irreversible measurable Finsler manifolds.
1.1.The result.The first result of this paper is the following.Theorem 1.3.Let (X, F, m) be a forward-complete measured Finsler manifold (possibly with boundary) satisfying the CD(0, N ) condition for some N > 1, having Euclidean volume growth.Then for every bounded Borel subset E ⊂ X it holds (1.7) P(E) ≥ N ω Moreover, inequality (1.7) is sharp.
As we already said, the possible irreversibility of the manifolds does not permit to simply apply Theorem 1.1 to Finsler manifolds.In order to prove Theorem 1.3, we will exploit the Brunn-Minkowski inequality which holds true also for Finsler manifolds.This strategy was used by Balogh and Kristály [7] for proving Theorem 1.1 and here we introduce no real new idea.Indeed, in the light of [29], it seems that this inequality holds true also for irreversible metric measure spaces; here we confine our-self to the setting of measured Finsler manifolds.To the best of the author knowledge, besides the Barky-Ledoux inequality [39], there is no other isoperimetric inequality for measured Finsler manifolds that does not involve the reversibility constant.
The main result of this paper concern the rigidity property of the isoperimetric inequality (1.7).To characterise its minima, we will have to additionally require the reversibility constant Λ F to be finite.This hypothesis is quite expected since in Finsler manifolds with infinite reversibility certain pathological behaviors may arise (e.g., the Sobolev spaces may not be vector spaces [28,22]).
Theorem 1.4.Let (X, F, m) be a forward-complete measured Finsler manifold (possibly with boundary) satisfying the CD(0, N ) condition for some N > 1, having reversibility constant Λ F < ∞.Assume that ∂X is locally forward convex (see Definition 2.4).Let E ⊂ X be a bounded Borel set that saturates (1.7).
Then there exists (a unique) o ∈ X such that, up to a negligible set, . Moreover, the measure m can be represented by the disintegration formula where m α is concentrated on the geodesic ray from o through α and m α can be identified (via the unitary speed parametrisation of the ray) with N ω N AVR X t N −1 L 1 [0,∞) .As an application of Theorem 1.4, we deduce the rigidity for the weighted anisotropic isoperimetric problem in Euclidean cones, in the irreversible case (the reversible case was already investigated [13]).We postpone this discussion to Section 1.2; now we briefly present the proof strategy of Theorem 1.4 and the structure of the paper.
The classical approach for proving a rigidity results consists in exploiting properties depending on the saturation of inequalities.In this paper, following the line of [13], we adopt a different approach that starts from the proof of the isoperimetric inequality for non-compact MCP(0, N ) spaces [12].In [13] it is used the localisation given by the L 1 -Optimal Transport problem between the renormalized measures restricted on the set E and B R , a large ball of radius R containing E. The localization gives a family of one-dimensional, disjoint transport rays together with a disintegration of the restriction to B R of reference measure m.At this point it is natural to analyze the well-known isoperimetric inequality for the traces of E along the one-dimensional transport rays.As Ohta pointed out [39], differently from the reversible case, the irreversibility of the space introduces the reversibility constant, obtaining a non-sharp inequality.Indeed, if one tries to prove Theorem 1.3 using the localization of E inside B R and taking the limit as R → ∞ (as it was done in [13,Theorem 4.3]) one would obtain a factor Λ −1 F in the lower bound.However, quite surprisingly, when studying the equality case, this factor will disappear.
In order to capture the equality it is therefore necessary to deal with this limit procedure and to get rid of the reversibility constant.The intuition suggests that, if E saturates inequality (1.7), then for large values of R the one-dimensional traces should be almost optimal.We intend the almost optimality in many respects: for example, the length of each transport ray has to be almost optimal; the disintegration measures has to have density t N −1 ; the traces of set E has to almost coincide with the interval starting from the starting point of the ray having as length the expected radius of ball saturating the inequality.This last observation will be the key-point for solving the issues arising by the irreversibility.
Indeed, we will see that the transport rays naturally come with a unitary vector field that "points outward" from the set E, and that, in the parametrization of the rays, this vector field points "to the right".When one computes the perimeter in the transport ray, one must compute the measure of the boundary of the trace of E; we divide the boundary in two parts: the part with outward normal vector pointing "to the right" and "to the left", respectively.For the former part one computes the measure as usual, while for the latter part one has to take into account the Finsler structure (here appears the reversibility constant).At this point the almost rigidity of the traces of E is used: the fact that any trace of E almost coincides with the interval [0, ρ] permits us to deduce that the part of the boundary "pointing to the left" contributes little to the perimeter, therefore we can get rid of the reversibility constant.
Having in mind these estimates we take the limit as R → ∞.There is no general procedure for taking the limit of a disintegration.However, following the procedure first employed in [13], we will exploit the almost optimality of the traces of E and the densities deduced in Section 6; these properties permit to obtain a well behaved limit disintegration for the reference measure restricted to the set E, as it described in Corollary 7.15.
Finally using the properties of the disintegration, we will deduce that the set E is a ball and the disintegration of the measure in the whole space (see Theorems 8.3 and 8.8, respectively).
The paper is organized as follows.Section 2 recalls a few facts on Finsler manifolds, the Curvature-Dimension condition and the localization technique.In Section 3 we prove Theorem 1.3.In Sections 4 and 5 we localize the reference measure and the perimeter and we present the one-dimensional reductions.In Section 6 the one-dimensional estimates are carried out.In Section 7 we deal with the limiting procedure, while Section 8 concludes proof of Theorem 1.4.We added two appendixes to this paper, containing the proof of the fact that the relative perimeter is a measure and that the perimeter is the relaxation of the Minkowski content.1.2.Applications in the Euclidean setting.As a consequence of Theorem 1.4, we present a characterization of minima for the weighted anisotropic isoperimetric problem in Euclidean cones.
The setting is the following: let Σ ⊂ R n be an open convex cone with vertex at the origin; let H : R n → [0, ∞) be a gauge, i.e., a non-negative, convex and positively 1-homogeneous function; let w : Σ → (0, ∞) be a continuous weight for the Lebesgue measure.
If E ⊂ R n is a set with smooth boundary, we define the weighted anisotropic perimeter relative to the cone Σ as (here ν and dS denote the unit outward normal vector and the surface measure, respectively).Under the assumptions that w 1 α is concave and w is positively α-homogeneous, it has been proven [33,11] the following sharp isoperimetric inequality for the weighted anisotropic perimeter (1.9) P w,H (E; Σ) , where N = n + α, W is the Wulff shape associated to H, and the expression w(A) with A ⊂ R n is a short-hand notation A w dx.
As observed in [11], Wulff balls W centered at the origin are always minimizers of (1.9).However in [11] the characterization of the equality case, is not present.Many efforts have been done for solving this problem.We now briefly recall the known results.For the unweighted case (w = 1), Ciraolo et al. [17] solved the problem under the assumption of H to be a uniformly elliptic positive gauge (i.e. a not necessarily reversible norm).The characterisation in weighted setting has been solved in [16] but only in the isotropic case (H = • 2 ).The author, together with Cavalletti, solved the weighted problem [13], with the assumption of H to be a norm (i.e.reversible) with strictly convex balls, knowing that the isoperimetric set is bounded.This paper improves [13] by dropping the reversibility assumption.
As it was observed in [11], the assumption that w 1/α is concave has a natural interpretation as the CD(0, N ) condition, where N = n + α.To be precise, if H is a gauge then its dual function F is a Finsler structure (with finite reversibility constant), provided that F is smooth outside the origin and F 2 is strictly convex (this two requirements can be equivalently required for the gauge H).One can associate to the triple Σ, H and w the measured Finsler manifold (Σ, F, wL n ).One can check that (Σ, F, wL d ) satisfies the CD(0, d + α) condition and that the unitary ball, if w ∈ C ∞ : in Chapter 10 of [40] and in [35] this is done in the case Σ = R n ; clearly the proof extends to the case of convex subsets.The fact that this manifold has finite reversibility, the convexity, and the forward-completeness are trivial checks.The perimeter associated to this space will indeed coincide with P w,H .Moreover, by the homogeneity properties of H and w, one can check that Indeed, recall that the Wulff shape W of H is the unitary ball of the Finsler structure F , hence the measure scales with power N = n + α.Conversely, the perimeter of the rescaled Wulff shape turns out to be the derivative w.r.t. the scaling factor of the measure, therefore the perimeter of the Wulff shape is N times its measure.This consideration shows that (1.9) follows from (1.7), thus we can apply Theorem 1.4 to (Σ, F, wL n ), obtaining the following result.
Theorem 1.5.Let Σ ⊂ R n be an open convex cone with vertex at the origin, and H : R n → [0, ∞) be a gauge.Assume H to have strictly convex balls and to be smooth outside the origin.Consider moreover the α-homogeneous smooth weight w : Σ → [0, ∞) such that w 1/α is concave.Then the equality in (1.9) is attained if and only if E = W ∩ Σ, where W is a rescaled Wulff shape.
Acknowledgement.The author is thankful to Fabio Cavalletti, Alexandru Kristály, and Shin-ichi Ohta for their comments to this manuscript.The authors are also grateful to the referees for reading the manuscript and suggesting a number of improvements.

Preliminaries
In this section we recall the main constructions needed in the paper.In Section 2.1 we review the geometry of measured Finsler manifolds; in Section 2.2 the perimeter in measured Finsler manifolds; in Section 2.3 the Curvature-Dimension condition; finally in Section 2.4 the localization method.
2.1.Finsler geometry.We quickly recall the basic notions regarding measured Finsler manifolds.The reader should refer to [40] for more details.We adopt the convention that a manifold may have a boundary, unless otherwise stated.We require the boundary to be Lipschitz.Definition 2.1.Let X be a connected n-dimensional manifold.We say that a function , where 0 denotes the null section; (2) (Positive 1-homogeneity) For all c > 0, v ∈ T X, it holds that F (cv) = cF (v); (3) (Strong convexity) On each tangent space T x X, the function F 2 is strictly convex.
The reader should notice that in general F (v) = F (−v).This feature, known as irreversibility, is what precludes us from applying the theory of m.m.s.'s.We define the reversibility constant of a Finsler structure as or, in other words, Λ X,F ∈ [1, ∞] is the least constant Λ X,F ≥ 1 such that for all v ∈ T X, F (v) ≤ Λ X,F F (−v).Later we will restrict ourself to the family of Finsler structures with finite reversibility.
If no confusion arises, we shall write We define the speed of a C 1 curve η as F ( η).The notion of speed naturally induces a length functional Remark 2.2.We reassure the reader on the fact that the lack of symmetry of the distance does not harm most of the classical theory of metric spaces.Indeed, one can build g 1 and g 2 , two riemannian metric on T X, such that (2.5) Such metrics can be built on local charts and then glued together using a partition of the unity.Furthermore, such metrics can be chosen so that g 2 ≤ f g 1 , for some continuous function f : Using these metrics one can reobtain many classical results for free.In particular, we will make use of the Ascoli-Arzelà theorem, the fact that locally Lipschitz functions (as will be later introduced) are differentiable almost everywhere, and that locally Lipschitz functions with compact support are globally Lipschitz.
We define the forward and backward balls, respectively, as If A ⊂ X, we define its (forward) ε-enlargement to B + (A, ε) := x∈A B + (x, ε).We say that a set A is forward (resp.backward) bounded, if for some (hence any) x 0 ∈ X, there exists r > 0 such that A ⊂ B + (x 0 , r) (resp.A ⊂ B − (x 0 , r)).A we say that a set is bounded if it both backward and forward bounded.We denote by diam A := sup x,y∈A d(x, y) the diameter of a set; a set has finite diameter if and only if it is bounded.Definition 2.3.Let (X, F ) be a Finsler manifold, possibly with boundary.We say that it is forwardcomplete, if and only if, for all sequences (x k ) k ⊂ X satisfying the forward Cauchy condition In light of Hopf-Rinow theorem, forward-completeness of a Finsler manifold is equivalent to the compactness of closed forward balls and implies that given two points there exists a minimal geodesic (as defined in the next paragraph) joining these two points.In case of manifolds without boundary, forward-completeness is equivalent also to the definition of the exponential map on the whole tangent bundle.
A curve γ : [0, l] → X is called minimal geodesic if it minimizes the length and its speed is constant.We point out that, if γ : [0, l] → X is a minimal geodesic, in general the reverse curve t → γ l−t may fail to be a geodesic due to the possible irreversibility of the manifold.We will denote by Geo(X) the set of minimal geodesic with domain the interval [0, 1].Like in the reversible setting, if γ ∈ Geo(X) is a minimal geodesic, it holds that in this case the condition t ≤ s cannot be lifted.
Definition 2.4.Let (X, F ) be a Finsler manifold with boundary.We say that ∂X is locally forward convex, if and only if, for all points x, y ∈ X\∂X, and for all minimal geodesic γ connecting x to y, we have that γ does not touch the boundary.
Given a submanifold Y ⊂ X, we can identify the tangent bundle T Y as a subset of T X via the standard immersion.With this notation, we can restrict the Finsler structure F to T Y ; clearly, F | T Y satisfies Definition 2.1.Regarding the reversibility constant and the distance, one immediately sees that (2.9) Λ Y,F ≤ Λ X,F , and d X,F (x, y) ≤ d Y,F (x, y), ∀x, y ∈ Y.
We define the dual function Notice that, while we have that ω(v) ≤ F * (ω)F (v), the "reverse" inequality may not hold: ω(v) −F * (ω)F (v).We define the Legendre transform L : T * x X → T x X as L(ω) = v, where v ∈ T x X is the unique vector such that F (v) = F * (ω) and ω(v) = F (v) 2 (the uniqueness follows from the fact that F 2 is smooth and strictly convex).Given a differentiable function f : X → R, we define its gradient as ∇f (x) := L(df (x)).Please note that in general ∇(−f ) = −∇f .
We say that a function f : We point out that the first inequality in (2.11) follows from the second by swapping x with y.
The family of L-Lipschitz functions is stable by pointwise limits; the infimum or the supremum of L-Lipschitz functions is still L-Lipschitz.Moreover, by Ascoli-Arzelà theorem, the family of L-Lipschitz functions forms a compact set in the topology of local uniform convergence.If , where these two expressions have the meaning of the slope of f as seen as a function defined in Y and X, respectively.If f is differentiable at x ∈ X, the slope can be computed as |∂f |(x) = F * (−df (x)) = F (∇(−f )(x)), hence for locally Lipschitz functions |∂f | = F (∇(−f )) almost everywhere.Finally, we would like to endow a manifold with a measure.Differently from the Riemannian case, there is no canonical measure induced from the Finsler structure.On the other hand the theory of m.m.s.'s does not require any strong assumption on the reference measure and, a priori, this measure might have nothing to do with the Hausdorff measure.For this reason we will only require for the reference measure to have a smooth density when expressed in coordinates.We conclude this section with the definition of measured Finsler manifold.Definition 2.5.A triple (X, F, m) is called measured Finsler manifold, provided that X is a connected differential manifold (possibly with boundary) F is a Finsler structure on X and m is a positive smooth measure, i.e., given x 1 , . . ., x n local coordinates, we have that 2.2.Perimeter.Following the line traced in [34,2,3] we give the definition of (relative) perimeter for measured Finsler manifold.Given a Borel subset E ⊂ X and Ω open, the perimeter of E relative to Ω is denoted by P(E; Ω) and is defined as follows, (2.14) In the unweighted Riemannian setting, if E has smooth boundary, it is a standard fact that P(E; Ω) = H n−1 (Ω ∩ ∂E).We say that E ⊂ X has finite perimeter in X if P(E; X) < ∞.We recall also a few elementary properties of the perimeter functions: (a) (locality) P(E; Ω) = P(F ; Ω), whenever m((E △ F ) ∩ Ω) = 0; (b) (lower semicontinuity) the map E → P(E, Ω) is l.s.c. with respect to the L 1 loc (Ω) convergence.Please note that, due to the possible irreversibility of the Finsler structure, in general the complementation property does not hold.If E is a set of finite perimeter, then the set function A → P(E; A) is the restriction to open sets of a finite Borel measure P(E; •) in X (see Appendix A), defined by Sometimes, for ease of notation, we will write P(E) instead of P(E; X).
Give a subset E ⊂ X, we define its (forward) Minkowski content as It can be shown that the perimeter is the l.s.c.relaxation of the Minkowski content w.r.t. the L 1 distance of sets.The proof of this fact can be found in Appendix B.
2.3.Wasserstein distance and the Curvature-Dimension condition.Given a forward-complete measured Finsler manifold (X, F, m), by M + (X) and P(X) we denote the space of non-negative Borel measures on X and the space of probability measures respectively.For p ∈ [1, ∞) we will consider the space P p (X) ⊂ P(X) of the measures µ satisfying (2.16) for some (hence any) o ∈ X, i.e. µ has finite p-th moment.On the space P p (X) we define the L p -Wasserstein distance W p , by setting, for µ 0 , µ 1 ∈ P p (X), (2.17) The infimum is taken over all probability measure π ∈ P(X × X) with µ 0 and µ 1 as the first and the second marginal, i.e., (P 1 ) ♯ π = µ 0 , (P 2 ) ♯ π = µ 1 , where P i , i = 1, 2 denote the projection on the first (resp.second) factor.The infimum is attained and this minimizing problem is called Monge-Kantorovich problem.
We call a geodesic in the Wasserstein space (P p (X), W p ) any curve µ : [0, 1] → P p such that It can be shown that if µ 0 and µ 1 are absolutely continuous, there exists a unique geodesic connecting µ 0 to µ 1 .
The CD(K, N ) for condition for m.m.s.'s has been introduced in the seminal works of Sturm [45,44] and Lott-Villani [30], and later investigated in the realm of measured Finsler manifolds [36] (see also the survey [37]); here we briefly recall only the basics in the case K = 0, 1 < N < ∞.
We define the N -Rényi entropy as Definition 2.6 (CD(0, N )).Let (X, F, m) be a forward-complete measured Finsler manifold and let N ∈ [dim X, ∞).We say that (X, F, m) satisfies the CD(0, N ) condition if and only if the N ′ -Rényi entropy is convex along the geodesic of the Wasserstein space ∀N ′ ≥ N , that is, for all couples of absolutely continuous curves µ 0 , µ 1 ∈ P 2 (X), it holds that where (µ t ) t∈[0,1] is the unique geodesic connecting µ 0 to µ 1 .
If (X, g, h Vol g ) is a weighted riemannian manifold, one can introduce the N -Ricci tensor (as defined in (1.2)).It was proven [45,44,30] that a weighted complete riemannian manifold without boundary, satisfies the CD(0, N ) condition if and only if Ric N ≥ 0.
Similarly to the riemannian case, a notion of weighted N -Ricci curvature, still denoted by Ric N , has been introduced.Here we do not give the definition of Ric N , for it is quite lenghty and useless for our purposes.Ohta [35] proved that a measured Finsler manifold without boundary satisfies the CD(0, N ) condition if and only if Ric N ≥ 0. The possible presence of the boundary in the manifolds the present paper deals with does not harm the results of this paper; indeed, we rely only on the curvature dimension condition given by Definition 2.6 and never on Ric N .
Among many consequences of the CD(0, N ) condition, two are of our interest.One is the Brunn-Minkowski inequality (see e.g.[40,Theorem 18.8]).Given two measurable subsets A and B of a CD(0, N ) measured Finsler manifold (X, F, m) and t ∈ [0, 1], we define With this notation, we have the Brunn-Minkowski inequality The other property we are interested in is the Bishop-Gromov inequality that states for any fixed point x ∈ X.This inequality guarantees that the definition of asymptotic volume ratio (see (1.4)) is well posed.

2.4.
Localization.The localization method, also known as needle decomposition, is now a wellestablished technique for reducing high-dimensional problems to one-dimensional problems.
In the Euclidean setting it dates back to Payne and Weinberger [42], it has been later developed and popularised by Gromov and V. Milman [24], Lovász-Simonovits [31], and Kannan-Lovasz-Simonovits [26].Klartag [27] reinterpreted the localization method as a measure disintegration adapted to L 1 -Optimal-Transport, and extended it to weighted Riemannian manifolds satisfying CD(K, N ).Cavalletti and Mondino [14] have succeeded to generalise this technique to essentially non-branching m.m.s.'s verifying the CD(K, N ), condition with N ∈ (1, ∞), and later Otha [39] developed this method for the Finsler setting.Here we only report the case K = 0.
In his work, Ohta considered manifolds without boundary.However, his proof also work for manifolds with local forward convex boundary.
Consider a measured Finsler manifold (X, F, m) satisfying the CD(0, N ) condition and a function f ∈ L 1 (m) with finite first moment such that (2.24) The function f induces two absolutely continuous measures µ 0 = f + m and µ 1 = f − m.The wellestablished theory of L 1 -optimal transport [46] specify that the Monge-Kantorovich problem possess a strictly related dual problem, the so-called Kantorovich-Rubinstein dual problem: where the maximum is taken among all possible 1-Lipschitz functions ϕ.The problem clearly admits a (non-unique) solution ϕ; we will call ϕ Kantorovich potential for the problem.Using ϕ we can construct the set inducing a partial order relation.The maximal chains of this order relation turns out to be the image of curves of maximal slope for ϕ with unitary speed.To be more precise, we introduce the concept of transport curve: we say that a unitary speed geodesics γ : Dom(γ) ⊂ R → X is a non-degenerate transport curve, if its domain has at least two points, d dt ϕ(γ(t)) = −1, and γ cannot be extended to a larger domain.
We distinguish three possible cases, according to the number of non-degenerate transport curves passing through a given point x ∈ X.
• There is no non-degenerate transport curve passing through x.We denote by D the set of such points.The set D is generally large.
2 Please, notice that we use a different sign convention from [39,40].
• There is exactly 1 non-degenerate transport curve passing through x.Such points form the so-called transport set that will be denoted by T .A fundamental property of T is that the f is constantly 0 a.e. in X\T .• There are 2 or more non-degenerate transport curves passing through x.Such points are called branching points and the set that they form will be denoted by A. The set A turns out to be negligible.All these sets are σ-compact, hence Borel.In the sequel, we will also refer to the sets of forward (resp.backward) branching points, defined as On the transport set, we define the relation R given by By construction, R is an equivalent relation on T and the equivalence classes are precisely the images of the transport curves.One can chose Q ⊂ T a Borel section of the equivalence relation R (this choice is possible as it was shown in [8,Proposition 4.4]).Define the quotient map where α is the unique element of Q such that (x, α) ∈ R. We shall denote by (X α ) α∈Q , the equivalence classes for relation R, and we will call them transport rays.
The existence of a measurable section permits to construct g : Dom(g) ⊂ Q × [0, +∞) → T , a measurable parametrization of the transport rays.Fix α ∈ Q and take γ, a transport curve, such that inf(Dom(γ)) = 0. Then define g(α, t) := γ t , whenever t ∈ Dom(γ).We specify that this parametrization guarantees that f (g(α, 0)) ≥ 0. By continuity of g w.r.t. the variable t, we extend g, in order to map also the end-points of the rays X α ; the restriction of g to the set {(α, t) The transport rays naturally come with the structure of one-dimensional oriented manifold, with the orientation given by ∂ t g(α, t), the velocity of the parametrization.We endow X α with the Finsler structure given by the restriction of F to X α ; notice that F (∂ t g(α, t)) = 1.As we already pointed out, it holds that Given a finite measure q ∈ M + (Q), such that q ≪ Q # (m T ), the Disintegration Theorem applied to (T , B(T ), m T ), gives a disintegration of m T consistent with the partition of T given by the equivalence classes {X α } α∈Q of R: Note that such measure q can always be build, by taking the push-forward via Q of a suitable finite measure mutually absolutely continuous w.r.t.m T .We recall by disintegration, we mean a map Remark 2.7.We point out that the disintegration is unique for fixed q.That means that, if there is a family ( mα ) α satisfying the conditions above, then for q-a.e.α, m α = mα .If we change q with a different measure q, such that q = ρq, ρ > 0, then the map α → ρ(α) −1 m α still satisfies the conditions above, with q in place of q.
We endow the transport ray X α with the measure m α , making (X α , F, m α ) a one-dimensional oriented measured Finsler manifold.
Differently from the reversible case, it might happen that the transport rays fail to satisfy the CD(0, N ) condition.However, a bound from below on the Ricci curvature can be given in a certain sense.It can be proved that m α = (g(α, • )) # (h α L 1 (0,|Xα|) ), for a certain non-negative function h α .The function h α satisfies (h ) ′′ ≤ 0 in the distributional sense, i.e., the function h is concave.Here we can recognize the CD(0, N ) for weighted riemannian manifolds, namely that the space ([0, ) satisfies the CD(0, N ) condition.This fact leads us to the following definition.Definition 2.8.Let (X, F, m) be a measured Finsler manifold diffeomorphic to an interval, endowed with an orientation given by a vector field v, such that F (v) = 1.We say that (X, F, m) satisfies the oriented CD(0, N ) condition (N > 1), if the following happens.There exists g : Dom With this definition, clearly holds that the transport rays satisfy the oriented CD(0, N ) condition.For the reader used with the notion of N -Ricci curvature, we point out that the oriented CD(0, N ) condition is equivalent to the fact Ric N (∂ t g(α, t)) ≥ 0.
Finally, we point out that, as a consequence of the properties of the optimal transport, we can localize the constraint X f dm = 0, i.e. it holds that X f dm α = 0, for q-a.e.α ∈ Q.
We summarize this section in the following theorem.
Theorem 2.9.Let (X, F, m) be a measured Finsler manifold satisfying CD(0, N ), for some Then there exists a measurable subset T ⊂ X (transport set), a family {(X α , F, m α )} α∈Q of oriented one-dimensional submanifolds of X (transport rays), and a measurable function g : Dom(g) ⊂ Q × [0, ∞) such that the following happens.The function f is zero m-a.e. in X\T and m T can be disintegrated in the following way, Moreover, for q-a.e.α ∈ Q, the transport ray (X α , F, m α ) is parametrized by the unitary speed geodesic g(α, • ), it satisfies the oriented CD(0, N ) condition, and it holds that Furthermore, two distinct transport rays can only meet at their extremal points (having measure zero for m α ).
Remark 2.10.The construction of the needle decomposition depends only on the function ϕ, rather than the function f .Indeed, given a 1-Lipschitz function ϕ one can construct the needle decomposition and reobtain Theorem 2.9, without, of course, Equation (2.32), which now makes no sense.

Proof of the main inequality
We devout this section in proving Theorem 1.3.
Proof of Theorem 1.3.We will first prove that From the inequality above the thesis will immediately follow by Theorem B.5. Fix E ⊂ X bounded and , hence there exist x ∈ E and y ∈ B + (x 0 , R) so that d(x, z) = td(x, y).By triangular inequality we deduce that , proving the claim.We are in position to compute the Minkowski content .
By taking the limit as R → ∞, recalling the definition of AVR X , we conclude.

Localization of the measure and the perimeter
From now on we assume that every Finsler manifold is forward-complete, that it has finite reversibility constant, and local forward convexity.To prove Theorem 1.4 we consider the isoperimetric problem inside a ball with larger and larger radius.In order to apply the needle decomposition given by the Localization Theorem 2.9, one also needs in principle the balls to be convex.As in general balls fail to be convex, we will overcome this issue in the following way.
Given a bounded set E ⊂ X with 0 < m(E) < ∞, fix a point x 0 ∈ E and then consider R > 0 such that E ⊂ B R (hereinafter we will adopt the notation B R := B + (x 0 , R)).Consider then the following family of null-average functions, Clearly, f R falls in the hypothesis of Theorem 2.9.Thus we obtain a measurable subset T R ⊂ X (the transport set) and a family {(X α,R , F, m α,R )} α∈Q R of transport rays, so that the measure m T R can be disintegrated: where m α,R are probability densities supported on X α,R .Let g R (α, • ) : [0, |X α,R |] → X α,R be the unit speed parametrisation of the transport ray X α,R , in the direction given by the natural orientation of the disintegration ray X α,R .With this notation, it holds for some CD(0, N ) density h α,R .The localization of the zero mean implies that (see (2.32)) Unfortunately, the presence of the factor m α,R (B R ) in the r.h.s. of the equation does make the quantity m α,R independent of α, harming the localization approach.To get rid of this factor we proceed as follows.
We define T α,R to be the unique element of [0, The measurability in α of m α,R implies the same measurability for and consequently we deduce T α,R ≤ R+diam(E).We restrict m α,R to the ray X α,R := g R (α, [0, T α,R ]), having the disintegration formula where T R := ∪ α∈Q R X α,R .Using (4.1) and the fact that The disintegration (4.6) will be a useful localisation only if (E ∩ X α,R ) ⊂ X α,R ; in this case we have obtaining a localization constraint independent of α.To prove this inclusion we will impose that has unitary speed, we notice that where in the second inequality we have used that each starting point of the transport ray has to be inside E ⊂ B R/(4Λ F ) , being precisely where f R > 0. The inequality above yields g R (α, t) where in the second inequality we used that g R (α, 0), , as we desired.
We describe explicitly the measure q R in term of a push-forward via the quotient map We study now the relation between the perimeter and the disintegration of the measure (4.6).Let Ω ⊂ X be an open set and consider the relative perimeter P(E; Ω).Let u n ∈ Lip loc (Ω) be a sequence such that u n → 1 E in L 1 loc (Ω) and lim n→∞ Ω |∂u n | dm = P(E; Ω).Using the Fatou Lemma, we can compute where |∂ X α,R u| denotes the slope of the restriction of u to the transport ray X α,R and P X α,R the perimeter in the submanifold ( X α,R , F, m α,R ).
By arbitrariness of Ω, we deduce the following disintegration inequality (4.8) Next proposition summarizes this construction.
Proposition 4.1.Let (X, F, m) be a CD(0, N ) measured Finsler manifold with Λ F < ∞.Given any bounded set E ⊂ X with 0 < m(E) < ∞, fix any point x 0 ∈ E and then fix R > 0 such that Then there exists a Borel set T R ⊂ X, with E ⊂ T R and a disintegration formula Moreover, the transport ray ( X α,R , F, m α,R ) satisfies the oriented CD(0, N ) condition and |X α | ≤ R + diam(E).Furthermore, it holds true that (4.11) The rescaling introduced in Proposition 4.1 will be crucially used to obtain non-trivial limit estimates as R → ∞.

One-dimensional analysis
Proposition 4.1 is the first step to obtain from the optimality of a bounded set E an almost optimality of E ∩ X α,R .We now have to analyse in details the behaviour of the perimeter in onedimensional oriented measured Finsler manifolds.
We fix few notation and conventions.A one-dimensional oriented measured Finsler manifold can be identified with the manifold (I, F, m), where I ⊂ R is an interval.Without loss of generality we assume that the orientation is given by the coordinated vector field ∂ t on I. Since we are studying manifolds arising from the localization, we shall consider only Finsler structures that satisfy F (∂ t ) = 1.Thus, it is clear that the Finsler structure is completely determined by F (−∂ t ); for this reason, with a slight abuse of notation, we will denote by F , the real-valued function given by F (−∂ t ).With this convention, the reversibility constant turns out to be (5.1) .
When the interval has finite diameter, we will always assume that I = [0, D].Notice that D in general is not the diameter, for it may happen that d(D, 0) > d(0, D) = D; however, it holds that diam(I, F ) ≤ Λ I,F D.
If (I, F, m) satisfies the oriented CD(0, N ) condition, then it happens that m is absolutely continuous w.r.t. the Lebesgue measure L 1 and (5.2) (h in the sense of distributions, where h = dm dL 1 .We stress out that if (I, F, hL 1 I ) satisfies the oriented CD(0, N ) condition, then the reversible manifold (I, | • |, hL 1 I ) satisfies the CD(0, N ) condition.We will say that the function h itself satisfies the CD(0, N ) condition if (5.2) holds.
Given a function h : I → [0, ∞), we shall write m h := hL 1 I .If the interval I is compact, we will assume also that D 0 h = 1, unless otherwise specified.We also introduce the functions notice that from the CD(0, N ) condition, h > 0 over (0, D) making v h invertible and in turn the definition of r h well-posed.
We will denote by P F,h the perimeter in the measured Finsler manifold (I, F, hL 1 I ).If E ⊂ [0, D] is a set of finite perimeter, then it can be decomposed (up to a negligible set) in a family of disjoint intervals (5.4) and the union is at most countable.In this case, we have that the perimeter is given by the formula (5.5) From the equation above, we immediately deduce a lower bound on the perimeter (5.6) 5.1.Isoperimetric profile function.The isoperimetric inequality for CD(0, N ) manifolds with bounded diameter is given in terms of the isoperimetric problem in the so-called model spaces.Here we recall the basic notions.
For N > 1, D > 0, and, ξ ≥ 0, we consider the model space , where For the model spaces, we can easily compute the functions v N,D (ξ, (5.9) The isoperimetric profile function for the model spaces is given by the formula (5.10) The family of one-dimensional measured Finsler manifolds satisfying the CD(0, N ) condition and having Λ F = 1 coincides with the of family of weighted Riemannian manifolds.E. Milman [32] gave an explicit lower bound for the perimeter of subset of manifolds in this family with the additional constraint of having diameter bounded by some constant D > 0. In other words, Milman proved that given As immediate consequence, one obtains that if we drop the reversibility hypothesis, the lower bound of the perimeter must be divided by the reversibility constant.
The author and Cavalletti proved [13, Lemma 4.1] deduced the an expansion for the isoperimetric profile, as follows.
Lemma 5.1.Fix N > 1.Then, we have the following estimate for I N,D (5.12) The following corollary incorporates both the irreversible and reversible case.
Corollary 5.2.Fix N > 1.Then for all D ≥ D ′ > 0 and for all one-dimensional oriented measured Finsler manifolds ([0, D ′ ], F, hL 1 ) satisfying the oriented CD(0, N ) condition, it holds that (5.13) Remark 5.3.The lower bound in (5.13) is very rough for our purposes.If one attempted to prove the isoperimetric inequality (1.7), the inverse of the reversibility constant would appear in the lower bound.
The only reason why the factor Λ −1 F appears in (5.13), is that the part of the boundary where the external normal vector "points to the left" might be non-empty.Indeed, if E is of the form [0, b], then P F,h (E) = P | • |,h (E).We will see that the part of the boundary "pointing to the left" contributes little to the perimeter.
We give the definition of the residual of a set.This object quantifies, in a way that will be detailed in Section 6, how far away is a ray from the expected model space.
If v ∈ (0, 1/2), we define the D-residual of v as (5.16) Notice that in the definition of Res D h (v) there is no dependence on the Finsler structure F ; indeed, the definition of Res D h (v) is given in terms of the perimeter of [0, r h (v)], and the perimeter of this set in [0, D ′ ] does not capture the possible irreversibility the Finsler structure.Using the residual, Inequality 5.13 can be restated as (5.17) On the other hand, whenever the set E is of the form [0, r] we obtain a much refined estimate 5.2.One-dimensional reduction for the optimal region.We are ready to apply the definition of residual to the disintegration rays.In order to ease the notation, we let P α,R = P ( X α,R ,F, m α,R ) .The measure m α,R will be identified with the ray map g R (α, •) to h α,R L 1 , thus we define Res α,R := Res The good rays are those rays having small residual.We quantify their abundance.
Proposition 5.5.Assume that (X, F, m) is a CD(0, N ) measured Finsler manifold, such that AVR X > 0. If E ⊂ X is a bounded set attaining the identity in the inequality (1.7), then Proof.In order to check that the function α → Res α,R is integrable, it is enough to check that (Res α,R ) − , is integrable.This last fact derives from the isoperimetric inequality as stated in (5.17).We can now compute the integral in (5.19) (5.20) and the r.h.s.goes to 0, as R → ∞.
Corollary 5.6.Let (X, F, m) be a CD(0, N ) measured Finsler manifold, having AVR X > 0. Let E ⊂ X be a set saturating the isoperimetric inequality (1.7), then it holds true that

Analysis along the good rays
The last theorem asserts (in a very weak sense) that the residual, in the limit for R → ∞, must be non-positive.Moreover, the measure of the traces of E is m(E) m(B R ) , hence infinitesimal.For this reason, we now use the residual and the measure of the set to control the density h : [0, D ′ ] → R, proving that in case of small measure and residual, h is close to the model density Similarly we prove that the traces of E are closed to the optimal, i.e., a certain interval of the form [0, r].Remark 6.1.We will extensively use the Landau's "big-O" and "small-o" notation.If several variables appear, but only a few of them are converging, either the "big-O" or "small-o" could in principle depend on the non-converging variables.However, this is not the case.
To be precise, in our setting, the converging variables will be w → 0 and δ → 0. Conversely the "free" variables will be: 1) D, a bound on the length of the ray; 2) D ′ ∈ (0, D], the length of the ray; 3) ([0, D ′ ], F, h) a one-dimensional measured Finsler manifold satisfying the oriented CD(0, N ) condition (in practice, each transport ray); 4) E ⊂ [0, D ′ ] a set with measure m h (E) = w and residual Res D F,h (E) ≤ δ.
The estimates we will prove are infinitesimal expansions as w → 0 and δ → 0 and whenever a "big-O" or "small-O" appears, it has to be understood that it is uniform w.r.t. the "free" variable.Remark 6.2.An important point to remark is the fact that we consider only the case when E is "on the left", i.e., E ⊂ [0, L], with the tacit understanding that L ≪ D ′ .This is possible because the transport rays come from the Optimal Transport problem between the bounded set isoperimetric E and the ball B R 6.1.Almost rigidity of the set E and of the length of the ray.We start considering the special case when the set E of the form E = [0, r].In this case the Finsler structure plays no role, for the outer normal vector on the boundary of E points to the right.For this reason, we omit the proof of the following proposition, because it is exactly what is proven in Propositions 5.3 and 5.4 of [13].Proposition 6.3.Fix N > 1.Then, for w → 0 and δ → 0 it holds that where D ≥ D ′ > 0 and ([0, D ′ ], F, hL 1 ) is one-dimensional measured Finsler manifold satisfying the oriented CD(0, N ) condition such that Res D h (w) = Res D F,h ([0, r h (w)]) ≤ δ.We now drop the assumption E = [0, r].Up to a negligible set, it holds that E = i∈N (a i , b i ), where the intervals (a i , b i ) are far away from each other (i.e.b i < a j or b j < a i , for i = j).The boundedness of the original set of our isoperimetric problem, implies that E ⊂ [0, L], for some L > 0. Define b(E) := ess sup E ≤ L.
In the next proposition we prove that b(E) is in the essential boundary of E. Proof.Taking into account the definition of residual and the isoperimetric inequality (5.17), choosing δ ≤ 1, we can deduce that (6.5) If we choose w small enough, taking into account the hypothesis D ≥ 4LΛ, we deduce D ′ ≥ 2L Since E = i (a i , b i ) (up to a negligible set), our aim is to prove that there exists j such that a i < a j , for all i = j.In this case we set a = a j .Suppose on the contrary, that ∀j, ∃i = j such that a i > a j , hence there exists a sequence (i n ) n , so that (a in ) n is increasing, thus converging to some y ∈ (0, L].Recalling that F ≥ Λ −1 , we can compute the perimeter We integrate obtaining (6.7) The first factor in the r.h.s. of the estimate above is controlled just using the definition of residual (6.8) and, if m h (E) → 0 and Res D F,h (E) is bounded, then the term above goes to 0. Regarding the second factor, it sufficies to prove that b D ′ is bounded: If we put together this last two estimates, we deduce that the r.h.s. of (6.7) is infinitesimal as m h (E) → 0 and Res D F,h (E) → 0, obtaining a contradiction.This proposition guarantees the existence of a right-extremal connected component of the set E; this component is precisely the interval (a, b(E)).We will denote by a(E) the number a given by Proposition 6.4.Since our estimates are infinitesimal expansions in the limit as m h (E) → 0 and Res D F,h (E) → 0, we will always assume that m h (E) ≤ w and Res D F,h (E) ≤ δ, so that the expression a(E) makes sense.For the same reason, we will always assume that h is increasing in the interval [0, b(E)].
We now prove that this component (a(E), b(E)) tends to fill the set E, that b(E) converges as expected to Dm h (E) 1 N , and that the length of the ray tends to be maximal.Proposition 6.5.Fix N > 1, L > 0, and Λ ≥ 1.Then, for w → 0 and δ → 0 it holds that 1 ) is a one-dimensional measured Finsler manifold satisfying the oriented CD(0, N ) condition with Λ F ≤ Λ, and the set E ⊂ [0, L] satisfies m h (E) = w and The thesis follows from estimate (6.1).Part 2 Inequality (6.12).Since the density h is strictly increasing on [0, b(E)] and E ⊂ [0, b(E)] (up to a null measure set), it holds that r h (w) ≤ b(E) and (6.14) Estimate (6.3) concludes this part (6.15) D(w Part 3 Inequality (6.13).
First we prove that a(E) < r h (w), for w and δ small enough.Suppose on the contrary that a(E) ≥ r h (w), implying that h(a(E)) ≥ h(r h (w)), hence ).We deduce that (compare with (5.18)) If we take the limit as w → 0 and δ → 0 we obtain a contradiction.
Using the Bishop-Gromov inequality and the isoperimetric inequality (respectively), we get We put together the inequalities above obtaining a(E) ≤ r h (w)Λ where the estimate (6.2) was taken into account.Part 4 Inequality (6.11).Since (6.19) Combining the inequality above, the already-proven estimate (6.12), and the estimate (6.2), we reach the conclusion.

Almost rigidity of the density h.
In this section we prove that the density h converges uniformly to the density N x N −1 /D N .The bound from below is easy and follows from the Bishop-Gromov inequality.

By combining these to estimates we reach the thesis
The following corollary gives a lower boundary for the residual, under the hypothesis that the (positive part of the) residual is bounded from above, improving inequality (5.17 F,h (E) ≤ δ.Proof.By a direct computation, recalling estimates(6.23)and (6.12), we obtain In order to prove an upper bound for the density, we present the following, purely technical lemma.
Proof.The proof is by contradiction.Suppose that there exists ε > 0 and three sequences in (η n ) n , (t n ) n , and (s n ) n , such that η n → 0, f (t n , 0) ≤ f (s n , η n ), and t n − s n > ε.Up to a taking a subsequence, we can assume that t n → t and s n → s, hence 1 . This implies f (s n , η n ) → f (s, 0), yielding f (t, 0) ≤ f (s, 0).Since t → f (t, 0) is strictly increasing, we obtain t ≤ s ≤ t − ε, which is a contradiction.
We now obtain an upper bound for h in the interval [a(E), b(E)] going in the opposite direction of the Bishop-Gromov inequality.Proposition 6.9.Fix N > 1, L > 0, and Λ ≥ 1.Then, for w → 0 and δ → 0, it holds that where D ≥ 4LΛ, D ′ ∈ (0, D], ([0, D ′ ], F, hL 1 ) is a one-dimensional measured Finsler manifold satisfying the oriented CD(0, N ) condition, with Λ F ≤ Λ, and the set E ⊂ [0, L] satisfies m h (E) = w and In order to ease the notation, define (6.30) a The concavity of h where in the last inequality we used the Bishop-Gromov inequality written in the form We now estimate the terms N w bl N−1 and a N b N .Regarding the former, taking into account (6.12) and the isoperimetric inequality (5.14), we deduce Conversely, we estimate the latter term (recall (6.11) and (6.13)) Putting all the pieces together, we obtain (6.34) where f is the function of Lemma 6.8.Applying said Lemma we get If we explicit the definitions of k, l, and b, it turns out that the inequality above is precisely the thesis.

6.3.
Rescaling the diameter and renormalizing the measure.So far, we have obtained an estimate of the densities h and the set E. The presence of factor 1 D N in the estimate (6.23) suggests the need of a suitable rescaling to get a non-trivial limit estimate.We rescale the space by 1 b(E) and renormalize the measure by m h (E).
Fix k > 0 and define the rescaling transformation S k (x) = x/k.Given a density h : [0, D ′ ] → R and E ⊂ [0, L], we define (6.36) Clearly ν h,E ≪ L 1 , so we denote by hE : [0, 1] → R the Radon-Nikodym derivative The density hE can be explicitly computed (6.37) Notice that, since E could be disconnected, the indicator function in (6.37) prevents h 1 from being concave, i.e., hE possibly fails the oriented CD(0, N ) condition.However, in the limit, the CD(0, N ) condition reappears, as it is explicated by the following proposition.Proposition 6.10.Fix N > 1, L > 0, and Λ ≥ 1.Then, for w → 0 and δ → 0 it holds that where having used the estimate (6.23), with x = tb(E), in the first inequality and (6.12) in the second inequality.
Part 2 Estimate from below and t ≤ a(E) b(E) .In this case it may happen that t b(E) / ∈ E, so the best estimate from below is the non-negativity.For this reason, here we exploit the fact that the interval [0, a(E) b(E) ] is "short" and that t ≤ a(E) b(E) .A direct computation gives (recall (6.12) and (6.13)) (6.40) Part 3 Estimate from above and t ≥ a(E) b(E) .
Part 4 Estimate from above and t ≤ a(E) b(E) .Without loss of generality we can assume that a(E) ∈ E. Using the previous part we compute The following theorem summerizes the contents of this section.Notice that the function ω takes as argument the positive part of the residual and not the residual itself.Theorem 6.11.Fix N > 1, L > 0, and Λ ≥ 1.Then there exists a function ω : (0, ∞) × [0, ∞) → R, infinitesimal in 0, such that the following holds.For all D ≥ 4LΛ, D ′ ∈ (0, D), for all ([0, D ′ ], F, hL 1 ) one-dimensional measured Finsler manifold satisfying the oriented CD(0, N ) condition with Λ F ≤ Λ, and for all E ⊂ [0, L], it holds that b(E) − Dm h (E) (6.44)where b(E) = ess sup E and hE is the density of m

Passage to the limit as R → ∞
We now go back to the studying the identity case of the isoperimetric inequality.Fix E a bounded Borel with positive measure such that (7.1) where (X, F, m) is a CD(0, N ) measured Finsler manifold having AVR X > 0. We will use the notation introduced Section 4. Denote by ϕ R the 1-Lipschitz Kantorovich potential associated to If we add a constant to ϕ R , we still get a Kantorovich potential, so we can assume that the family ϕ R is equibounded on every bounded set.The Ascoli-Arzelà theorem, together with a diagonal argument, implies that that, up to subsequences, ϕ R converges to a certain 1-Lipschitz function ϕ ∞ , uniformly on every compact set.
We recall the disintegration given by Proposition 4.1 The effort of this section goes in the direction to understand how the properties of the disintegration behave at the limit, and to try to pass to the limit in the disintegration.Throughout this section, we set ρ = ( m(E) ω N AVR X ) 1 N .Before going on, using the self-improvement estimate of the residual (6.44), we prove the following Proposition.
Proposition 7.1.Up to taking subsequences, it holds that Using estimate (6.44), we estimate the negative part of the residual where ω is a function, infinitesimal in (0, 0).The L 1 -norm of the residual is given by Taking into account the previous inequality and, again, Corollary 5.6, we deduce that Res Q R (x),R , converges to 0 in L 1 .By taking a subsequence, we obtain (7.3).
7.1.Passage to the limit of the radius.First of all we define the radius function r R : ), where the notation b(E) was introduced in Section 6.1.
The radius function is defined on E for two motivations: we require a common domain not depending on R and the domain must be compact.
Remark 7.2.The set E ∩ T R has full m E -measure in E, hence it does not really matter how r R is defined outside E ∩ T R .This fact is relevant, because we will only take limits in the m E -a.e.sense.
The next proposition ensures that, in limit as R → ∞, the function r R converges to ρ, which is precisely the radius that we expect.Proposition 7.3.Up to subsequences it holds true Proof.By Proposition 7.1, there exists a sequence R n and a negligible subset N ⊂ E, such that lim n→∞ Res Q Rn (x),Rn = 0, for all x ∈ E\N .Define G := n T Rn \N and notice that m(E\G) = 0. Fix n ∈ N and x ∈ G and let α : The second term goes to 0 by definition of AVR, so we focus on the first term.Consider the ray ( X α,Rn , F, m α,Rn ).By definition, we have that (7.9) Res Rn+diam E h α,Rn We can now use Theorem 6.11 (in particular estimate (6.42)), obtaining (7.10) Taking the limit as n → ∞, we conclude.
7.2.Passage to the limit of the rays.Consider now a constant-speed parametrization of the rays inside the set E: where x ∈ E and s ∈ [0, 1].Remark 7.2 applies also to the map x → γ x,R .A direct consequence of the definition of γ x,R and the properties of the disintegration are x ∈ E, (7.12) for m-a.e.
The next proposition guarantees that the properties (7.17)-(7.19)pass to the limit as R → ∞.
This means that the equation above holds true except for a set of measure at most ε, and by letting ε → 0, we conclude.Finally we prove (7.22).In this case, consider the continuous, non-negative function L(x, γ) := inf t∈[0,1] d(x, e t (γ)).Equation (7.19) implies The equation above passes to the limit as R → ∞, so the conclusion immediately follows.
7.3.Disintegration of the measure and the perimeter.Having in mind the disintegration formula (7.2), we define the map E ∋ x → µ x,R ∈ P(E) as A direct computation (recall (4.9)-(4.10))gives thus the following disintegration formula holds, (7.27) Remark 7.5.We briefly discuss the measurablity of the integrand function in Eq. (7.27).It holds that the map x → µ x,R (A) is measurable and the formula (7.27) holds.Indeed, the map x → µ x,R (A) is (up to excluding the a negligible set) the composition of the maps ), we can compute explicitly the measure µ x,R (recall that by (7.7) r R (x) = ess sup E x,R , for m E -a.e.x) (7.28) Having in mind (4.11), we can perform a similar operation for the perimeter.Indeed, in the natural parametrization of the rays, if we consider only the "right extremal" of E x,R and the fact that This observation, naturally leads to the definition Using the maps γ x,R and hx,R , we rewrite p x,R By definition of p x,R we have that (7.32) x ∈ E, deducing the following "disintegration" formula (equations (4.11) and (4.10) are taken into account), ( Define now the compact set F := e (0,1) (K) = {γ t : γ ∈ K, t ∈ [0, 1]} and let S ⊂ M + (F ) be the subset of the non-negative measures on F with mass at most N/ρ.The sets P(F ) and S are naturally endowed with the weak topology of measures.Since K and F are compact Hausdorff spaces, the Riesz-Markov Representation Theorem, implies that the weak topology on P(F ) and S, coincides with the weak* topology induced by the duality against continuous functions C(K) and C(F ), respectively.The weak* convergence can be metrized on bounded sets, if the primal space is separable; here we chose as a metric where {f k } k is a dense set in C(X).We endow the spaces P(F ) and S with the distance defined in (7.34), making them two compact metric spaces.Define now the map G R : E × K → P(F ) × S, as Clearly, the function G R is measurable w.r.t. the variable x and continuous w.r.t. the variable γ.Define the measure (having mass m(E)) In order to ease the notation, we set Z = E × K × P(F ) × S.
Taking into account we continue this chain of inequalities Since P(E; A) = inf{P(E; Ω) : Ω ⊃ A is open}, for any Borel set A, we can conclude.
Before taking the limit as R → ∞, we state a useful lemma.
Lemma 7.7.Let X be a Polish space, let Y , Z be two compact metric spaces, and let m be a finite Radon measure on X.Consider a sequence of functions f n : X × Y → Z and f : X × Y → Z, such that f and f n are Borel-measurable in the first variable and continuous in the second.Suppose that for m-a.e.x ∈ X the sequence f n (x, •) converges uniformly to f (x, •).Consider a sequence of measures Then we have that Proof.In order to ease the notation, set ν n = (Id × f n ) # µ n and ν = (Id × f ) # µ.Fix ε > 0. We make use an extension of the Egorov's and Lusin's Theorems for functions taking values in separable metric spaces (see [21,Theorem 7.5.1]and [20,Appendix D]).In this setting, we deal with maps taking value in C(Y, Z), the space of continuous functions between the compact spaces Y and Z, which is separable.Therefore, there exists a compact K ⊂ X such that: 1) the maps x ∈ K → f n (x, •) ∈ C(Y, Z) are continuous; 2) the restrictions x ∈ K → f n (x, •) converge to x ∈ K → f (x, •), uniformly in the space C(K, C(Y, Z)); 3) m(X\K) ≤ ε.Regarding point 2), this implies that the restrictions We test the convergence of ν n against a function Regarding the second term, we compute the integral (7.42) Using compactness, one easily checks that ϕ(x, y, f n (x, y)) converges to ϕ(x, y, f (x, y)) uniformly in K × Y .For this reason, together with the fact that µ n ⇀ µ weakly we take the limit in the equation above concluding the proof and let σ := (Id ×G) # τ .Then it holds that σ R ⇀ σ in the weak topology of measures.
Proof.We need only to check the hypotheses of the previous Lemma.Due to Remark 2.2, the irreversiblity of the distance is not harmful.The set E is compact, hence Polish.The set K is compact and so is P(F ) × S (w.r.t. the distance given by (7.34)).The maps G R are measurable and continuous in the first and second variable, respectively.Finally, we need to see that for m E -a.e.x, the limit G R (x, γ) → G(x, γ) holds uniformly in γ.Fix x and γ and pick ψ ∈ C b (F ) a test function and compute The r.h.s. of the inequality is independent of γ (but depends only on x and ψ) and converges to 0 by Theorem 6.11, (see particular (6.43)).Therefore, the first component of G R (x, γ) converges (in the weak topology of P(F )), uniformly w.r.t.γ (compare with (7.34)).For the other component the proof is analogous, so we omit it.
We conclude this section with a proposition reporting all the relevant properties of the limit measure σ.
Proposition 7.9.The measure σ satisfies the following disintegration formulae x ∈ γ, (7.48)We prove now (7.44).Given a function ψ ∈ C 0 b (F ) = C 0 b (e (0,1) (K)) we define L ψ : P(F ) → R as L ψ (µ) = F ψ dµ.This latter function is bounded and continuous w.r.t. the weak topology of P(F ), thus we can compute the limit using (7.37) and ( 7 Using standard approximation arguments, we see that the equation above holds true also for any ψ ∈ L 1 (E; m E ).
If we test the inequality above with ψ = 1, the inequality is saturated thus the two measures have the same mass, so the inequality improves to an equality.
7.4.Back to the classical localization notation.We are now in position to re-obtain a "classical" disintegration formula for the measure m, as well as for the relative perimeter of E.
We recall the definition of some of the objects that were introduced in Section 2.4.For instance, let Γ ∞ := {(x, y) : ϕ ∞ (x) − ϕ ∞ (y) = d(x, y)} and let T ∞ be the transport set, i.e., the family of points passing through only one non-degenerate transport curve.Let A ∞ the set of branching points (i.e. points where two of more non-degenerate transport curves pass).The sets of forward and backward branching points are defined as the quotient map; denote by X α,∞ := Q −1 (α) the disintegration rays and let g ∞ : Dom(g ∞ ) ⊂ Q ∞ × [0, ∞) → X be the standard parametrization of the rays.
We introduce the function t α : X α,∞ → [0, ∞) defined as (7.53) the function t α measures how much a point is translates from the starting point of the ray X α,∞ .
The next corollary concludes the discussion of the limiting procedures of the disintegration.

E is a ball
The aim of this section is to prove that E coincides with a ball of radius ρ and to extend the disintegration formula to the whole manifold.Before starting the proof, we give a topological technical lemma.This lemma is, in some sense, a weak formulation of the statement: let Ω be an open connected subset of a topological space X and let E ⊂ X be any set; if Ω ∩ E = ∅ and Ω\E = ∅, then we have that ∂E ∩ Ω = ∅.Lemma 8.1.Let (X, F, m) be measured Finsler manifold (with possible infinite reversibility).Let E ⊂ X be a Borel set and let Ω ⊂ X be an open connected set with finite measure.If m(E ∩ Ω) > 0 and m(Ω\E) > 0, then P(E; Ω) > 0.
Proof.Assume first that the manifold is riemannian.In this case, we can assume by contradiction that P(E; Ω) = 0, yielding that the BV function 1 E is constant in Ω, but this contradicts the hypotheses m(E ∩ Ω) > 0 and m(Ω\E) > 0.
We now drop the riemannianity hypothesis.As we stressed out (see Remark 2.2), there exists a riemannian metric g, such that its dual metric g −1 in T * X satisfies g −1 (ω, ω) ≤ F * (ω), for all ω ∈ T * X.By definition of perimeter, there exists a sequence in Ω, we conclude that P (X,g,m) (E; Ω) ≤ P (X,F,m) (E; Ω).Proposition 8.2.For q-a.e.α ∈ Q ∞ , it holds that Proof.We prove only the former inequality; the latter has the same proof.In order to ease the notation define Clearly, n ≪ m (compare with (7.71)), thus ϕ ∞ (x) ≤ M , for n-a.e.x ∈ E. If we compute the integral we can deduce that q(H) = 0 and, by arbitrariness of ε, we conclude.
Theorem 8.3.There exists a (unique) point o ∈ X, such that, up to a negligible set, Proof and the equality of measures improves to an equality of sets.
By lipschitz-continuity of ϕ ∞ we deduce Continuing the chain of inequalities, we arrive at The line above, together with the fact that q(H) > 0, contradicts Proposition 8.2.
Finally, using again (8.Proof.If x ∈ E ∩ T ∞ , then x = g(α, t), for some t, with α = Q ∞ (x).By the previous proposition we may assume that g ∞ (α, 0) = o, hence we have that Since T ∞ ∩ E has full measure in B + (o, ρ), we conclude.
8.2.Localization of the whole space.We can now extend the localization given in Section 7.4 to the whole space X.Since we do not know the behaviour of ϕ ∞ outside B + (o, ρ), we take as reference 1-Lipschitz function −d(o, • ), which coincides with ϕ ∞ on B + (o, ρ): we disintegrate using −d(o, • ) and we see that this disintegration coincides with the one given Section 7.4 in the set E. From this fact, and the geometric properties of the space, we will conclude.We recall some of the concepts introduced in Subsection 2.4, applied to the 1-Lipschitz function −d(o, • ).The set D where no non degenerate transport curve pass is empty, for we can connect o to any point with a minimal geodesic.The set of branching points, A, contains only o and elements of the boundary; this follows from the uniqueness of the geodesics.For this reason, the transport set T coincides with X\{o}.Let Q ⊂ T be a measurable section and let Q : T → Q be the quotient map; let X α := Q −1 (α) be the disintegration rays and let g : Dom(g) ⊂ Q × R → X be the standard parametrization.The map t → g(α, t) is the unitary speed parametrization of the geodesic connecting o to α and then maximally extended.Define q := 1 m(E) Q # (m E ).Using the CD(0, N ) condition, one immediately sees that Q # (m) ≪ q.
Proof.Fix ε > 0 and let (8.20) with the convention that the limit above is 0 if |X α | < ∞.The limit always exists and it is not larger than ω N AVR X by the Bishop-Gromov inequality applied to each transport ray.We compute AVR X using the disintegration = ω N AVR X (1 − εq(C)), thus q(C) = 0.By arbitrariness of ε we deduce that lim R→∞ R 0 h α /R N = ω N AVR X , hence h α (t) = N ω N AVR X t N −1 , for q-a.e.α ∈ Q.
The proof of Theorem 1.4 is therefore concluded.As described in the introduction, Theorem 1.5 is an immediate consequence.Proof.We consider the case m + (E) < ∞ (the other is trivial).This implies that m(E\E) = 0, hence, without loss of generality, we may assume that E is closed.Consider the ε −1 -Lipschitz function Clearly f ε → 1 E in L 1 (m).In B + (E, ε 2 ) it is equal to 1, hence |∂f ε |(x) = 0, for all x ∈ E. Conversely, in X\B + (E, ε + ε 2 ) it attains its minimum, hence |∂f ε |(x) = 0 for all x ∈ X\B + (E, ε + ε 2 ).We compute the integral By taking the inferior limit as ε → 0, we conclude.
The previous proposition guarantees that the l.s.c.envelope of the Minkowski content is not smaller than the perimeter.The reverse is a bit more difficult and, at a certain point, we will require forwardcompleteness.
We consider the "semigroup" (T t ) t≥0 given by the formula (B.3) T t f (x) := sup If x is a point where f is differentiable, then the last term of the inequality above is equal to F * (−df ) = |∂f |(x), concluding the proof.
We prove now a sort of coarea formula.Proof.In the first place, we notice that ∞ 0 1 {f ≥t} (x) dt = f (x).Fix t ≥ 0 and h > 0. If x ∈ B + ({f ≥ t}, h), then (T h f )(x) ≥ t, or in other words 1 B + ({f ≥t},h) ≤ 1 {(T h f )≥t} .By integrating over t we obtain (B.7) By subtracting the first equation to the inequality above, integrating over x and using Fubini's theorem, we obtain The set {f ≥ 0} is compact, hence for h > 0 sufficiently small B + ({f ≥ 0}, h) is compact.Moreover, is smaller than the Lipschitz constant of f , hence the integrand in the r.h.s. is dominated by an L 1 function.We take the inferior and superior limit in the l.h.s. and r.h.s.(respectively) of the inequality above; the Fatou's Lemma brings us to the conclusion.
We now prove that we can, without loss of generality, assume that the functions of a sequence attaining the minimum in the definition of the perimeter have compact support.
Proposition B.4.Let (X, F, m) be a forward-complete measured Finsler manifold, and let E ⊂ X be a Borel set with finite measure.Then there exists a sequence of Lipschitz functions with compact support, (w n ) n , such that w n → 1 E in L 1 and P(E) = lim n→∞ X |∂w n | dm.
Proof.Fix E ⊂ X with finite measure, such that P(E) < ∞ (otherwise the proof is trivial).Let

F
( η) dt, and thus we have a natural notion of distance between two points given by (2.3) d X,F (x, y) := inf η {Length(η) : η 0 = x, and η 1 = y}.Whenever no confusion arises, we shall write d = d X,F .The distance d satisfies the usual properties of a distance, with the exception of the symmetry (2.4) d(x, y) ≤ d(x, z) + d(z, y), ∀x, y, z ∈ X, and d(x, y) = 0 ⇔ x = y.

Lemma 6 . 4 .
Fix N > 1, L > 0, and Λ ≥ 1.Then there exists two constants w > 0 and δ > 0 (depending only on N , L, and Λ) such that the following happens.For all D ≥ D ′ > 0 with D ≥ 4LΛ, for all ([0, D ′ ], F, hL 1 ) a one-dimensional measured Finsler manifold satisfying the oriented CD(0, N ) condition with Λ F ≤ Λ, and for all E ⊂ [0, L], such that m h (E) ≤ w and Res D F,h (E) ≤ δ, there exists a ∈ [0, b(E)) and an at-most-countable family of intervals ((a i , b i )) i such that, up to a negligible set,E = i (a i , b i ) ∪ (a, b(E)), (6.4) with a i , b i < a, ∀i.Moreover, h is strictly increasing on [0, b(E)].

Proposition B. 1 .
Let (X, F, m) be a measured Finsler manifold and E ⊂ X be a Borel set.Then it holds that (B.1) m + (E) ≥ P(E).

Lemma B. 2 .
y∈B − (x,t) f (y), T 0 f = f.Note that the ball in the supremum is backward.The semigroup T t enjoys the following immediate property.It holds that T t+s f ≥ T t (T s f ) and, if f is locally Lipschitz(B.4)lim sup t→0 + T t f − f t ≤ |∂f |, m-a.e. in X.Proof.Regarding the first part, fix x ∈ X, and ε > 0. By definition there exists y such that d(y, x) < t and (T t (T s f ))(x) ≤ (T s f )(y)+ε.Similarly, there exists z such that d(z, y) < s and (T s f )(y) ≤ f (z)+ε.By triangular inequality, we have that d(z, x) < t + s, thus(B.5)(T t+s f )(x) ≥ f (z) ≥ (T s f )(y) − ε ≥ (T t (T s f ))(x) − 2ε.By arbitrariness of ε, we conclude the first part.Regarding the second part, fix x ∈ X.By a direct computation we deduce lim supt→0 + (T t f )(x) − f (x) t = inf r>0 sup t∈(0,r) sup y∈B − (x,t) f (y) − f (x) y) − f (x)) + d(y, x)= lim sup y→x (f (y) − f (x)) + d(y, x) .
γ, µ, p) ∈ Z : e ε (γ) ∈ A + ∞ and conditions (7.44)-(7.50)holds} Notice that by definition of A + ∞ , if (x, γ, µ, p) ∈ P , then γ t ∈ A + ∞ , for all t ∈ [0, ε], thus we can compute 0 . Define Ẽ := supp 1 E .Recall that by definition of support, Ẽ = C C, where the intersection is taken among all closed sets C such that m(E\C) = 0; and in particular m(E\ Ẽ) = 0. Let o ∈ arg max Ẽ ϕ ∞ .By definition of Ẽ, we have that max Ẽ ϕ ∞ = ess sup E ϕ ∞ , deducing the first equality of (8.3).The other equality in (8.3) will follow from the fact E = B + (o, ρ) (up to a negligible set).It is sufficient to prove only that B + (o, ρ) ⊂ E, for the other inclusion is automatic Indeed, the Bishop-Gromov inequality, together with the definition of a.v.r.yields