Equality in Borell-Brascamp-Lieb inequalities on curved spaces

By using optimal mass transportation and a quantitative H\"older inequality, we provide estimates for the Borell-Brascamp-Lieb deficit on complete Riemannian manifolds. Accordingly, equality cases in Borell-Brascamp-Lieb inequalities (including Brunn-Minkowski and Pr\'ekopa-Leindler inequalities) are characterized in terms of the optimal transport map between suitable marginal probability measures. These results provide several qualitative applications both in the flat and non-flat frameworks. In particular, by using Caffarelli's regularity result for the Monge-Amp\`ere equation, we {give a new proof} of Dubuc's characterization of the equality in Borell-Brascamp-Lieb inequalities in the Euclidean setting. When the $n$-dimensional Riemannian manifold has Ricci curvature ${\rm Ric}(M)\geq (n-1)k$ for some $k\in \mathbb R$, it turns out that equality in the Borell-Brascamp-Lieb inequality is expected only when a particular region of the manifold between the marginal supports has constant sectional curvature $k$. A precise characterization is provided for the equality in the Lott-Sturm-Villani-type distorted Brunn-Minkowski inequality on Riemannian manifolds. Related results for (not necessarily reversible) Finsler manifolds are also presented.

Here, for every s ∈ (0, 1), p ∈ R ∪ {±∞} and a, b ≥ 0, the p-mean is defined by Characterizations of cases of equality and the problem of stability in the aforementioned inequalities (1.2)-(1.4) are still subjects for further investigation. After the pioneering works by Brunn and Minkowski, it is well known for more than a century that equality in (1.3) holds if and only if the sets A and B are homothetic convex bodies from which sets of measure zero have been removed; similarly, equality in (1.4) holds if and only if the sets A and B are translated convex bodies up to a null measure set. The equality case in the generic Borell-Brascamp-Lieb inequality (1.2) has been studied in the mid of seventies by Dubuc [17] on R n by using deep convexity and measure theoretical results together with a careful inductive argument w.r.t. the dimension of the space R n . Later on, Dancs and Uhrin [15,16] obtained some qualitative Borell-Brascamp-Lieb inequalities on R, providing also some higher-dimensional versions. A few years ago, Ball and Böröczky [2,3] obtained stability results for the one-dimensional functional Prékopa-Leindler inequality with some extensions also to higher-dimensions. Very recently, various stability results are established in R n for the generic Borell-Brascamp-Lieb inequality by Ghilli and Salani [24], Rossi and Salani [36], for the Prékopa-Leindler inequality by Bucur and Fragalà [8], and for the Brunn-Minkowski inequality by Christ [10], Figalli and Jerison [18][19][20] and Figalli, Maggi and Pratelli [21,22]. The common strategy in the aforementioned papers, up to the latter two papers, is the use of various arguments from convex analysis combined usually with some inductive step w.r.t. the dimension, by fully exploring the Euclidean character of the space. In [21,22], quantitative Brunn-Minkowski inequalities are established by using optimal mass transportation arguments in R n . Further results concerning equality and stability in the Brunn-Minkowski inequality in R n can be found in Milman and Rotem [31] and Colesanti, Livshyts and Marsiglietti [12].
As far as we know, no equality/stability results are available for Borell-Brascamp-Lieb inequalities on curved spaces. It is clear that the arguments from the aforementioned papers (see [2], [3], [8], [10], [15], [17], [18], [19], and references therein) cannot be applied in such a nonlinear setting. The starting point of our investigation is the celebrated work by Cordero-Erausquin, McCann and Schmuckenschläger [14] who established a Riemannian version of the Borell-Brascamp-Lieb inequality via optimal mass transportation culminating in a distorted Jacobian determinant inequality. The Finslerian counterparts of the results from [14] are provided by Ohta [33]. We point out that the first optimal mass transportation approaches to geometric inequalities have been provided by Gromov in [32] (via the Knothe map) and McCann [29], [30,Appendix D] (via the Brenier map).
The main purpose of our paper is to characterize the equality in Borell-Brascamp-Lieb inequalities on complete n-dimensional Riemannian/Finsler manifolds for the whole spectrum of the parameter p ≥ − 1 n by exploring a quantitative Hölder inequality and the theory of optimal mass transportation. In the sequel, we roughly present some of our achievements. , g(y) v s (x, y) for all (x, y) ∈ M × M, z ∈ Z s (x, y), (1.5) where v s is the volume distortion coefficient (see (3.1) for its precise definition). Under the assumption (1.5), the main result of Cordero-Erausquin, McCann and Schmuckenschläger [14] says that where the integrals are considered w.r.t. the canonical volume element dV w on (M, w). For simplicity of notation, let · 1 be the L 1 -norm of any integrable function on M. For the above functions f, g and h, let us consider the Borell-Brascamp-Lieb deficit given by We first provide an estimate for the Borell-Brascamp-Lieb deficit on a general Riemannian manifold that will be achieved by using optimal mass transportation and a quantitative Hölder inequality: Theorem 1.1. (Estimate of the Borell-Brascamp-Lieb deficit) Let (M, w) be a complete ndimensional Riemannian manifold, s ∈ (0, 1), p ≥ − 1 n and f, g, h : M → [0, ∞) be three nonzero, compactly supported integrable functions satisfying (1.5). Then where ψ : M → M is the unique optimal transport map from the measure µ =f dV w to ν =gdV w with densitiesf = f / f 1 ,g = g/ g 1 , and G p,n s ≥ 0 is the gap-function given in Lemma 2.1.
The uniqueness of the optimal transport map ψ : M → M from the probability measure µ =f dV w to ν =gdV w is well known by McCann [28] having the form ψ(x) = exp x (−∇ϕ(x)) for a.e.
x ∈ supp f , and Jac(ψ s )(x) its Jacobian in a.e. x ∈ supp f . By Theorem 1.1 the equality in the Borell-Brascamp-Lieb inequality can be characterized by studying the properties of the gap-function G p,n s , leading us to the following result: be a complete n-dimensional Riemannian manifold, s ∈ (0, 1), p > − 1 n and f, g, h : M → [0, ∞) be three nonzero, compactly supported integrable functions satisfying (1.5). Then the following two assertions are equivalent: (a) δ p M,s (f, g, h) = 0, i.e., equality holds in the Borell-Brascamp-Lieb inequality; (b) the following statements simultaneously hold: (i) supp h = ψ s (supp f ) up to a null measure set; The equality in the Borell-Brascamp-Lieb inequality for p = − 1 n is treated separately in Theorem 3.1. We point out that usually the inclusion ψ s (supp f ) ⊂ supp h is strict. According to Theorem 1.2(b), the equality in the Borell-Brascamp-Lieb inequality implies the equality ψ s (supp f ) = supp h, which corresponds in R n to the Alesker-Dar-Milman parametrization of the Minkowski sum of two sets; for further details, see Remark 4.2. Theorem 1.2 provides both well known and genuinely new rigidity results; we briefly present some of them in the sequel (for details, see §4): • Equality in the Borell-Brascamp-Lieb inequality in R n : a new approach to Dubuc's characterization. As a first consequence of Theorem 1.2 we prove that equality in the Borell-Brascamp-Lieb inequality in R n holds if and only if the functions f, g and h are obtained as compositions of fixed (t, p)-concave function Φ with appropriate homotheties, where the support of Φ is convex up to a null set; for the precise statement, see Theorem 4.1. This result provides a new qualitative formulation of Dubuc's characterization, see [17,Théorème 12]. Our strategy relies on applying Theorem 1.2 in order to reduce the problem to the equality case in the Brunn-Minkowski inequality for the marginal supports supp f and supp g, implying the convexity of these sets. Using the convexity of the support of the target measure, a suitable application of the celebrated regularity result of Caffarelli [9] provides smoothness of the optimal mass transport map which turns to be an affine function in R n . We notice that Caffarelli's regularity has been already employed in order to establish sharp stability results in R n for the Brunn-Minkowski inequality (1.3), see Figalli, Maggi and Pratelli [21,22].
• Equality in Borell-Brascamp-Lieb inequality implies constant curvature. We state that the equality in the Borell-Brascamp-Lieb inequality on an n-dimensional Riemannian manifold with Ricci curvature Ric(M ) ≥ k(n−1) for some k ∈ R can be expected to hold only when a particular region of the manifold between the marginal supports has constant sectional curvature k; see Theorem 4.2 for details. The proof is based on Theorem 1.2 and a careful comparison argumentà la Bishop-Crittenden of the volume distortion coefficients with suitable quantities involving Jacobi fields on space forms.
• Equality in distorted Brunn-Minkowski inequalityà la Lott-Sturm-Villani. For some for s ∈ (0, 1), k ∈ R and n ≥ 2, let be the distortion coefficient introduced independently by Lott and Villani [26] and Sturm [39] in order to define their famous curvature-dimension condition CD(k, n) on metric measure spaces. Let (M, w) be a complete n-dimensional Riemannian manifold with Ricci curvature bounded below, i.e., Ric(M ) ≥ k(n − 1) for some k ∈ R (which is equivalent to the validity of CD(k, n)) and let us denote by m the canonical measure on M w.r.t. the volume element dV w . The distorted Brunn-Minkowski inequality reads as where A, B ⊂ M are measurable sets with m(A) = 0 = m(B) and • Negatively curved case: if k < 0 and (M, g) has nonpositive, nonzero sectional curvature (e.g., the hyperbolic space H n ), equality in (1.6) cannot hold for any positive measure sets A and B. The proof of the latter statements are based on a porosity argument and a geometric form of the Steinhaus density theorem (concerning the 'difference' of two sets).
The organization of the paper is as follows. In Section 2 we provide the proof of the quantitative Hölder inequality which is crucial in the proof of Theorem 1.1. In Section 3 we prove simultaneously Theorems 1.1 and 1.2. Section 4 is devoted to rigidity results. Accordingly, in §4.1 we prove Theorem 4.1 which provides a qualitative version of Dubuc's characterization. In §4.2 we deal with Riemannian manifolds by proving that the equality in Borell-Brescamp-Lieb inequality implies constant curvature, see Theorem 4.2, and we discuss the equality cases in the distorted Brunn-Minkowski inequality (1.6), see Theorem 4.3. In §4.3 certain Borell-Brascamp-Lieb inequalities are presented on not necessarily reversible Finsler manifolds, highlighting some subtle differences between Riemannian/Euclidean and Finslerian frameworks, respectively.

A quantitative Hölder inequality
According to Gardner [ for every a, b, c, d ≥ 0, s ∈ (0, 1) and p, q ∈ R such that p + q ≥ 0 with η = pq p+q when p and q are not both zero, and η = 0 if p = q = 0.
In the sequel, we provide a technical improvement of (2.1) needed to prove Theorem 1.1.   Proof. We first recall the quantitative Young inequality, i.e., if r ≥ 2 and 1 r + 1 r = 1, one has 2) see e.g. Cianchi [11].
(i) Let p ∈ (0, ∞) and let us assume first that pn ≥ 1. Applying inequality (2.2) for r = p p = pn+1 ≥ 2 and r = 1 pn , we have that  Since rp = p and the desired relation follows. The case pn ≤ 1 follows in the same way. If p ∈ (− 1 n , 0). Since −p = − p pn+1 ∈ (0, ∞) and p =p pn+1 , we can apply the previous estimate by reversing the roles of the means.
(ii) We first assume that s ≥ 1 2 , i.e., min(s, 1 − s) = 1 − s. We apply (2.2) with r = 1 1−s ≥ 2 and r = 1 s , obtaining Rearranging the above inequality, and using M 0 s (ac, bd) ≥ M − 1 n s (ac, bd) and the Bernoulli inequality, it follows that If s ≤ 1 2 , we proceed in a similar way as above. (iii) We first assume that a ≥ b. Then we have After a rearrangement, Bernoulli's inequality and (2.1) give the required inequality. The same can be done for a ≤ b.
The proof of (iv) directly follows by (iii); we left it to the interested reader.

The Borell-Brascamp-Lieb deficit: proof of main results
Let (M, w) be a complete n-dimensional Riemannian manifold and d : M × M → R be its distance function. Let B(x, r) = {y ∈ M : d(x, y) < r} be the geodesic ball with center x ∈ M and radius r > 0. Fix s ∈ (0, 1). According to Cordero-Erausquin, McCann and Schmuckenschläger [14], the volume distortion coefficient in where m is the Riemannian measure given by m(S) = S dV w for every measurable set S ⊂ M and dV w is the canonical Riemannian volume form.
The proof of Theorems 1.1 and 1.2 will be presented simultaneously.
Proof of Theorems 1.1 and 1.2. We recall that ψ s : M → M is the s−interpolant optimal transport map given by ψ s (x) = exp x (−s∇ϕ(x)) for a.e. x ∈ supp f for some d 2 /2-concave function ϕ : M → R.
We first notice that up to a null measure set. Indeed, if x ∈ A, then ψ(x) ∈ supp g and by the hypothesis (1.5) and convention on M p s , it follows that h(ψ s (x)) > 0. Furthermore, we also have the injectivity of the interpolant ψ s on A, see [14,Lemma 5.3].
In the proof we consider several cases according to the values of p.
and there are equalities in the above estimates. In particular, up to a null measure set of M , which gives property (i) of Theorem 1.2.
, the latter relation is equivalent to for a.e. x ∈ A.
By (1.5) and the above estimate we necessarily have for a.e. x ∈ A that , which is (iii) of Theorem 1.2 . Since we also have equality in the Jacobi determinant inequality (3.2), property (ii) of Theorem 1.2 directly follows by (iii); thus every item of (b) holds true. The reverse implication is trivial.
Case 2: p = +∞. A similar reasoning as in Case 1 and Lemma 2.1 (iii) give that for a.e. x ∈ A.
Furthermore, in order to have the equality case, by (1.5) and the latter relation we necessarily have for a.e. x ∈ A that which corresponds to (iii) of Theorem 1.2. Clearly, one also has (i) and by the equality in (3.2) we necessarily have for a.e. x ∈ A that which is precisely (ii) of Theorem 1.2. The converse is trivial again. Case 3: p = 0. Similarly as above, by Lemma 2.1 (ii) we have Let us assume that δ 0 M,s (f, g, h) = 0; thus, the latter integrand is zero. Note that G 0,n s (a, b, c, d) = 0 if and only if ac = bd. Therefore, we obtaiñ for a.e. x ∈ A.
Having equality in (1.5), from the latter relation we obtain for a.e. x ∈ A that which is (iii) of Theorem 1.2. Property (i) follows trivially, while (ii) comes from (iii) and the equality in (3.2), i.e., Case 4: p = − 1 n . The proof is similar to the case p = +∞; indeed, one has By Lemma 2.1 (iv), the claim follows. The equality case is treated in the following result. (i) supp h = ψ s (supp f ) up to a null measure set; M,s (f, g, h) = 0, it follows by Case 4 of the previous proof that f 1 = g 1 . Clearly, (i) holds true again by Case 4. Finally, (ii) follows by direct computation.

Applications: rigidity results
Our main results (Theorems 1.1 and 1.2) can be efficiently applied to establish various rigidity results. In §4.1 we consider the Euclidean case, in §4.2 the case of Riemannian manifolds, while in §4.3 we discuss the case of Finsler manifolds. The notations are kept from the previous sections.

4.1.
Euclidean case: Dubuc's result recovered via optimal mass transportation. Let t ∈ (0, 1) and p ∈ R ∪ {+∞}. We say that a nonnegative integrable function Φ : In such a case, the latter notation is simply called p-concavity, see Gardner [23,Section 9]. In particular, in the latter case, the p−concavity The main result of this subsection provides a novel, qualitative characterization of the equality case in the Borell-Brascamp-Lieb inequality, complementing the result of Dubuc [17] (see also Rossi and Salani [36]): (a) δ p R n ,s (f, g, h) = 0, i.e., equality holds in the Borell-Brascamp-Lieb inequality (1.2); (b) there exist an element x 0 ∈ R n , a convex set K ⊂ R n with K = supp f up to a null measure set and a (t, p)−concave function Φ : K → R with t = sc 0 1−s+sc 0 and c 0 = L n (supp g) L n (supp f ) 1 n such that up to null measure sets and for a.e. x ∈ K, Hereafter, the following two conventions are used: • if p = 0 then it will turn out by the proof that c 0 = 1, thus we may consider c Proof of Theorem 4.1. (a) =⇒ (b) We distinguish two cases. Case 1: p > − 1 n . Taking into consideration that in the Euclidean case the distortion coefficients v s (x, y) are identically equal to 1, according to Theorem 1.2, the equality in the Borell-Brascamp-Lieb inequality, i.e., δ p R n ,s (f, g, h) = 0, is characterized by: For simplicity, let A = supp f and B = ψ(A). We also recall that ψ : A → B is the optimal transport map from the measure µ =f dL n to ν =gdL n , wheref = f / f 1 andg = g/ g 1 It is clear by (1.1) (or (1.5)) and the definition of M p s that . By a change of variables and (ii), it follows that On the other hand, by the Monge-Ampère equation (3.3) forf andg, one hasf (x) =g(ψ(x))Jac(ψ)(x) for a.e. x ∈ A; in particular, by the last relation of (iii) we have that Therefore, by (4.3) one has Combining the above two relations, we obtain that i.e., we have equality in the Brunn-Minkowski inequality. By Gardner [23, p. 363], we know that A and B are homothetic convex bodies from which sets of measure zero are removed. Let K and S be these convex bodies (which differ in null sets by A and B, respectively), and let c 0 > 0 and  The latter relation and the strict concavity of det 1 n (·) over the cone of nonnegative definite symmetric matrices give that Hessη(x) = c 0 I n for every x ∈ K, where (4.4) Therefore, Accordingly, by (iii) we have that (4.5) Now, let x 1 , x 2 ∈ K be arbitrarily fixed elements. Let y 2 := ψ(x 2 ) = c 0 x 2 + x 0 ∈ S. By (4.5) we have that Replacing now the above relations into (1.1) for x 1 and y 2 , it follows that M p pn+1 s 1, , (4.6) We distinguish two cases: Case 1a: p = 0. Note that by (4.4) one has c 0 = 1 and relation (4.6) reduces to e., f is a (s, 0)-concave function in K.
Case 1b: p = 0. Again by (4.4), a simple computation and relation (4.6) give that i.e., f is a (t, p)-concave function in K with t = sc 0 1−s+sc 0 . The rest of the proof of (4.2) follows by (4.5).
To treat this case, we recall the inequality where f 1 , f 2 : A → R are nonnegative, integrable functions on a measurable set A ⊂ R. The proof of (4.7) follows by the Newton binomial expansion and the classical Hölder inequality for integrals; moreover, equality in (4.7) holds if and only if for some c > 0 we have f 2 (x) = cf 1 (x) for a.e. x ∈ A.
Due to Theorem 3.1, the equality in the Borell-Brascamp-Lieb inequality, i.e., δ − 1 n R n ,s (f, g, h) = 0, is characterized by: (i) supp h = ψ s (supp f ) up to a null measure set; Let us keep the previous notations, i.e., A = supp f , B = ψ(A) and the convex function η : R n → R with ψ = ∇η. By (ii) we have that f (x) = h(ψ s (x))Jac(ψ s )(x) for a.e. x ∈ A, thus f 1 = h 1 . Moreover, by (iii) and the Monge-Ampère equation (3.3) it follows that f (x) = g(ψ(x))Jac(ψ)(x) for a.e. x ∈ A. In particular, ψ s (A)). Therefore, Consequently, in the latter estimates we necessarily have equalities. First, being equality in the Brunn-Minkowski inequality, the sets A and B are homothetic convex bodies up to a null measure set; let K and S be the convex bodies which differ in null measure sets by A and B, respectively, and c 0 > 0 and x 0 ∈ R n such that S = c 0 K + x 0 . Second, by the equality case in (4.7), we have for some c > 0 that f (x) g(ψ(x)) = c for a.e. x ∈ A. In particular, by (ii) we have It is clear that c = L n (B) L n (A) = c n 0 . By the convexity of S, relation (4.9) and the regularity result of Caffarelli [9], it turns out that η is of class C 2 on K. Furthermore, by (4.9) we have thus the strict concavity of det 1 n (·) on the cone of nonnegative definite symmetric matrices implies that Hessη(x) = c 0 I n for every x ∈ K. Accordingly, Therefore, by (ii) we have Let x 1 , x 2 ∈ K be two arbitrarily fixed elements. Let y 2 := ψ(x 2 ) = c 0 x 2 + x 0 ∈ S. By (4.10) we have that g( which is equivalent to which means that f is (t, − 1 n )-concave in K with t = sc 0 1−s+sc 0 . The relations for g and h from (4.2) easily follow by (4.10).
Although our approach is more appropriate for characterizing equality cases, we conclude the present subsection by stating weak stability results for Brunn-Minkowski-type inequalities, e.g. for the log-Brunn-Minkowski inequality (1.4); an exhaustive study of the latter inequality can be found in Böröczky, Lutwak, Yang and Zhang [7].  On the other hand, for a.e. x ∈ A, we have x ∈ A, which implies that c 0 = 1. The converse is trivial.
By taking the limit r → 0, one may choose s k (0) = 1.
x ∈ supp f = A one has: (i) the sectional curvature is equal to the constant k along the geodesic t Proof. Since Ric(M ) ≥ (n − 1)k, Bishop's comparison principle implies that for every x ∈ M , y ∈ M \ cut(x) and s ∈ (0, 1), v s (x, y) ≥ s k (sd(x, y)) s k (d(x, y)) . (4.14) On the other hand, by relations (4.12)-(4.14) and the monotonicity of M p s (·, ·) we have for a.e. x ∈ A that Consequently, we have equalities in the above estimates. Again, by the monotonicity of M p s (·, ·) we necessarily have for a.e. x ∈ A that which proves (ii)&(iii) through Theorem 1.2 (b)(iii) and Theorem 3.1, respectively. If Y (s) = d(exp x ) −s∇ϕ(x) denotes the Jacobian of the exponential map at −s∇ϕ(x) ∈ T x M, relation (4.15) implies in particular that for a.e.
Due to Bishop and Crittenden [5, §11.10], an analysis of the behavior of Jacobian fields shows that for a.e. x ∈ A the sectional curvature along the geodesics t → ψ t (x), t ∈ [0, 1] is constant, having the value k, which concludes the proof of (i).  Proof of Theorem 4.3. Let us suppose that A and B are two compact subsets of M and s ∈ (0, 1) such that equality holds in (1.6). As we shall see, the most difficult part will be to prove the statement about the geodesic convexity of A and B in part (i). By monotonicity reasons it turns out that (4.12) holds. Therefore, due to Theorem 4.2, one has δ +∞ M,s (f, g, h) ≥ 0, which is precisely the generalized Brunn-Minkowski inequality (1.6).
In the sequel, let us assume that we have equality in (1.6), i.e., Moreover, by Theorem 4.2 (ii), we also have for a.e. x ∈ A that  be the t 0 −neighborhood of B. It is clear that A ∩ B t 0 = ∅. Indeed, if we assume that x ∈ A ∩ B t 0 , then there exists y ∈ B such that d(x, y) < t 0 , which contradicts the fact that t 0 = Θ A,B . Now, let us fix x ∈ A such that d(x, ψ(x)) = t 0 ; due to (4.18), the latter happens for a.e. x ∈ A. By construction, we have that B(ψ(x), t 0 ) ⊂ intB t 0 = B t 0 , thus B(ψ(x), t 0 ) ∩ A = ∅. Fix r 0 ∈ (0, t 0 ). Then, for every 0 < r < r 0 let us fix z r ∈ Z r 2t 0 (x, ψ(x)); then B(z r , r 2 ) ⊂ B(x, r) ∩ B(ψ(x), t 0 ). Therefore, B(z r , r 2 ) ⊂ B(x, r) \ A, i.e., A is 1 2 -porous at x, see Rajala [35]. In particular, the set A has null measure, m(A) = 0, which contradicts our assumption, proving the first part of the assertion. Now, we assume the sets A and B do not contain cut locus pairs and there is equality in (1.6). By Cases 1&2 we know that A, B and Z s (A, B) coincide up to a null measure set. Accordingly, without loss of generality we may consider the case that Z s (A, A) = A ∪ C where m(C) = 0. The proof of the geodesic convexity of A (up to a null measure set) is divided into several steps.
To prove this, let us first observe that A * ⊆ A ⊆ A, where A denotes the set of accumulation points of A. Indeed, the first inclusion follows by the definition of the density set A * , while the latter comes from the closeness of A.
Let x, y ∈ A * be arbitrarily fixed; we shall prove that {z} = Z s (x, y) ⊂ A. Note that z ∈ Z s (x, y) is unique since x / ∈ cut(y). Moreover, the latter fact also implies that there are neighborhoods U and V of x and y, respectively, such that x / ∈ cut(y ) for every (x , y ) ∈ U × V . Clearly, we may choose Since Z s (A, A) = A ∪ C with m(C) = 0, the estimate (4.19) shows that Z s (A m x , A m y ) contains a positively measured subset of A. Therefore, for every m ∈ N large enough, let us choose such a triplet (x m , y m , z m ) with x m ∈ A m x , y m ∈ A m y and {z m } = Z s (x m , y m ) ⊂ A; the element z m is also uniquely determined since x m / ∈ cut(y m ). We shall prove that the sequence (z m ) m converges to z (up to a subsequence) and z ∈ A. Since M is compact (following by the Bonnet-Myers theorem) and z m ∈ A for every m ∈ N, there existsz ∈ M such that lim m→∞ z m =z ∈ A ⊂ A. It remains to prove thatz = z. By Z s (x m , y m ) = {z m }, we have that d(x m , z m ) = sd(x m , y m ) and d(z m , y m ) = (1 − s)d(x m , y m ). Taking the limit as m → ∞, it follows that d(x,z) = sd(x, y) and d(z, y) = (1 − s)d(x, y), i.e.z ∈ Z s (x, y). By uniqueness, we havẽ z = z, which concludes the proof of Claim 1.
Claim 2: A * is open. This statement can be seen as a curved version of the Steinhaus theorem, see [38]. First, let us observe that (A * ) * = A * . Indeed, since A * ⊆ A, the inclusion (A * ) * ⊆ A * is trivial. Conversely, if we assume that there exists x ∈ A * \ (A * ) * , it follows that for every ε > 0 sufficiently small there exists r ε > 0 such that for every 0 < r < r ε we have m(A ∩ B(x, r)) ≥ (1 − ε)m(B(x, r)) and m(A * ∩ B(x, r)) ≤ (η + ε)m(B(x, r)) for some η ∈ [0, 1). Therefore, one has Let p ∈ A * = (A * ) * and fix r > 0 such that B(p, 2r) is a totally normal neighborhood of p. First, let us assume that 1 2 ≤ s < 1. We introduce the function R p : B(p, r) → B(p, r) which associates to each x ∈ B(p, r) the point R p (x) by reflecting x through p such that p ∈ Z s (x, R p (x)). We notice that R p (x) ∈ B(p, r) (since 1 2 ≤ s < 1) and the point R p (x) is uniquely determined, i.e., R p is well defined. Fix 0 < δ < r sufficiently small that will specified later; performing the same construction for every q ∈ B(p, δ) instead of p, we defined the function R q : B(p, r) → B(p, r + δ) such that q ∈ Z s (x, R q (x)) for all x ∈ B(p, r). (4.20) Since p ∈ (A * ) * , for every ε > 0 sufficiently small there exists r ε > 0 such that for every 0 < r < r ε , we have m(A * ∩ B(p, r)) ≥ (1 − ε)m(B(p, r)). By the Borel regularity of the measure m one can find a compact set K ⊂ A * ∩ B(p, r) such that m(K) ≥ (1 − 2ε)m(B(p, r)). Now, choose δ < r so small that R q (K) ⊂ B(p, r) and m(R q (K)) ≥ 1 − s 2s m(K) for all q ∈ B(p, δ). The inclusion R q (K) ⊂ B(p, r) follows by a continuity reason. In order to verify the inequality in (4.21), let us observe first that R q (x) = exp q •R • exp −1 q (x), x ∈ B(p, r), where R : T q M → T q M is the s-reflection given by R(y) = − 1−s s y, y ∈ T q M. Since exp q is a diffeomorphism on B(p, r) and d(exp q ) 0 = id, the map exp q is a local bi-Lipschitz map with bi-Lipschitz constant arbitrarily close to 1, which concludes the proof of (4.21).
With this choice of δ > 0, we shall prove that B(p, δ) ⊂ A. By contradiction, let us assume that there exists q ∈ B(p, δ) such that q / ∈ A. We notice that there is no x ∈ K such that R q (x) ∈ K. Indeed, by contrary, we would have that x ∈ A * and R q (x) ∈ A * , thus by (4.20) and Claim 1 we get q ∈ Z s (x, R q (x)) ⊂ Z s (A * , A * ) ⊆ A, which contradicts q / ∈ A. Therefore, for every x ∈ K one has that R q (x) / ∈ K, i.e., K ∩ R q (K) = ∅. On the other hand, since K ∪ R q (K) ⊆ B(p, r), by (4.21) we have (B(p, r)), a contradiction. Accordingly, B(p, δ) ⊂ A. Since B(p, δ) is open, one has that B(p, δ) = B(p, δ) * ⊆ A * . The case 0 < s < 1 2 works similarly by interchanging (s, 1 − s) with (1 − s, s). Accordingly, the function R p : B(p, r) → B(p, r) will be defined by reflecting x through p with the property that p ∈ Z 1−s (x, R p (x)) = Z s (R p (x), x) (instead of p ∈ Z s (x, R p (x))); the same should be performed in (4.20) for R q , q ∈ B(p, δ), i.e., q ∈ Z s (R q (x), x). x (y)) ∈ U . Accordingly, by Claim 1 one has Z s (A * , A * ) = Z s (A * , A * ) * ⊆ A * . Claim 4: A * is geodesic convex. Let x, y ∈ A * (x = y), and d 0 := d(x, y). Since x / ∈ cut(y), let γ : [0, 1] → M be the unique minimal geodesic joining x and y, parametrized proportionally to arc-length. Since A * is open, there exists δ > 0 such that B(x, δ) ∪ B(y, δ) ⊂ A * . If δ ≥ d 0 , we have nothing to prove, since Im(γ) ⊂ B(x, δ) ⊂ A * . If δ < d 0 , let s 0 < δ and let I = B(x, s 0 ) ∩ Im(γ) and J = B(y, s 0 ) ∩ Im(γ) be two geodesic segments in γ with lengths s 0 , i.e., Hereafter, B(x, r) = {y ∈ M : d(x, y) ≤ r}, r > 0. By the minimality of γ, we clearly have that Z s (I, J) ⊂ Im(γ); more precisely, by the parametrization we have that Z s (I, and its length is s 0 . Moreover, since I ∪ J ⊂ A * , by Claim 3 we also have that Z s (I, J) ⊆ A * . Repeating this argument, we cover the whole geodesic segment γ after finitely many steps with such pieces of geodesic segments of length s 0 , all of them belonging to A * .
(ii) (Negatively curved case) Due to (1.7), one has Θ A,B = max{d(x, y) : x ∈ A, y ∈ B} > 0. Similarly as above, relation (4.17) implies that The proof is 'dual' to (i); for completeness, we provide it. Let Since According to (4.22), for a.e. x ∈ A, one has d(x, ψ(x)) = t 0 and x / ∈ cut(ψ(x)); let us choose such an x ∈ A. It is clear that M \ B(ψ(x), t 0 ) ⊂ intB t 0 = B t 0 , thus (M \ B(ψ(x), t 0 )) ∩ A = ∅. Since x / ∈ cut(ψ(x)), we may extend the minimal geodesic joining the point ψ(x) to x beyond x such that the extended geodesic is still minimizing between ψ(x) and points in a small neighborhood of x. Let z r ∈ M be such a point belonging to the extended geodesic with d(z r , x) = r 2 for sufficiently small r > 0; thus, d(z r , ψ(x)) = d(z r , x) + d(x, ψ(x)) = r 2 + t 0 . This construction shows that B(z r , r 2 ) ⊂ B(x, r) and B(z r , r 2 ) ⊂ M \ B(ψ(x), t 0 ), i.e., B(z r , r 2 ) ⊂ B(x, r) \ A, which means that A is 1 2 -porous at x. Consequently, one has m(A) = 0, which contradicts our assumption.
(iii) (Null curved case) Let π :M → M be the universal covering of M , see Boothby [6,Corollary 9.8]. We consider the pull-back metric onM such that π becomes a local isometry.
Let (4.23) In particular, Z s (A, B) = ψ s (A) up to a null measure set and by Theorem 4.2(i) we have that for a.e.
Some further remarks are in order after the proof of Theorem 4.3.
where Θ min A,B = min{d(x, y) : x ∈ A, y ∈ B} and Θ max A,B = max{d(x, y) : x ∈ A, y ∈ B}. The proof of (i) in Theorem 4.3 treats actually the equality case at the left hand side of (4.26). Roughly speaking, when A and B are two disjoint positive measure sets, such an equality does not hold since the target measure ν cannot be reached by push-forwarding the measure µ; the transport cost (Θ min A,B ) 2 is not enough to realize this transportation. A similar explanation works also in the 'dual' case (ii); here, the equality in the distorted Brunn-Minkowski inequality corresponds to the equality at the right hand side of (4.26). In this setting, such an equality cannot be realized since by push-forwarding the measure µ to ν the transport cost (Θ max A,B ) 2 is too large; in fact, either we transport (a positive mass of) µ beyond ν, or we use non-optimal paths to reach ν from µ. For p = +∞ the latter inequality reduces to the distorted Brunn-Minkowski inequality (1.6).
We conclude this subsection by characterizing the equality in Brunn-Minkowski inequality via the flatness of the manifold; namely, we have Proof. (i) Assume that we have equality in (4.27) for every geodesic balls A = B(x, r) and B = B(y, R) with x, y ∈ M and r, R > 0 sufficiently small. Let ψ : A → B be the optimal transport map from the measure µ = 1 A /m(A)dV w to ν = 1 B /m(B)dV w . By Theorem 4.3(iii), the sectional curvature is zero along the geodesics t → ψ t (x), t ∈ [0, 1], joining a.e. x ∈ A to ψ(x) ∈ B. The arbitrariness of the sets A and B and a density argument shows that the sectional curvature on (M, w) is zero.
(ii) If (M, w) is isometric to R n , we have equality in (4.27) for every balls. Conversely, if (M, w) is simply connected, the equality case in (4.27) for geodesic balls implies that (M, w) has zero sectional curvature (from (i)). By the Killing-Hopf theorem it follows that (M, w) is isometric to R n . • Riemannian manifold, whenever g ij (x) = g ij (x, y) is independent of y.
• locally Minkowski space, if there exists a local coordinate system (x i ) on M with induced tangent space coordinates (y i ) such that F depends only on y = y i ∂/∂x i and not on x. • Minkowski space, whenever M is a finite dimensional vector space (identified by R n ) which is endowed by a Minkowski norm, inducing a Finsler metric on R n by translations. • Berwald space, whenever the coefficients Γ k ij (x, y) of the Chern connection are independent of y. It is clear that Riemannian manifolds and (locally) Minkowski spaces are Berwald spaces. For further concepts and results from Finsler geometry we refer to Bao, Chern and Shen [4], Kristály [25], Ohta [33] and Shen [37].
Given µ and ν two absolutely continuous measures on (M, F ) w.r.t. the normalized volume form dV F with compact support, there exists a unique optimal transport map from µ to ν of the form ψ(x) = exp x (∇(−ϕ(x))), where ϕ : M → R is a d 2 F /2-concave function on M, see Ohta [33,Theorem 4.10]. Here, d F : M × M → R is the usual Finsler metric, and ∇ is the Finslerian gradient. For s ∈ (0, 1) fixed, let ψ s (x) = exp x (s∇(−ϕ(x))) be the s-intermediate optimal transport map. The key tool to prove Borell-Brescamp-Lieb inequalities on Finsler manifolds is the Jacobian inequality where Jac(ψ s )(x) and Jac(ψ)(x) are the Jacobian determinant of ψ s and ψ at x and for all (x, y) ∈ A × B, z ∈ Z s (x, y). Then δ p M,s (f, g, h) ≥ 0. Moreover, if δ p M,s (f, g, h) = 0 then for a.e.
x ∈ supp f = A, one has (i) the flag curvature is equal to the constant k along the geodesic t → ψ t (x), t ∈ [0, 1], for flags having the form {S, v} with S = span{u, v} ⊂ T ψt(x) M and v = d ψ(x)), then f 1 = g 1 and  Example 4.1. On R n−1 (n ≥ 2) we introduce a complete Riemannian metric w such that (R n−1 , w) has nonnegative Ricci curvature, and for every ε ≥ 0, we define on R n = R n−1 × R the metric F ε : T R n = R 2n → [0, ∞) for every (x, t) ∈ R n and (y, v) ∈ T x R n−1 × T t R = R n by F ε ((x, t), (y, v)) = w x (y, y) + v 2 + ε w x (y, y) 2 + v 4 .
(R n , F ε ) is a Riemannian manifold if and only if ε = 0; however, if ε > 0, then (R n , F ε ) is a noncompact, complete, reversible non-Riemannian Berwald space with nonnegative Ricci curvature.
Fix ε > 0. According to Corollary 4.2, if equality holds in (4.30) for arbitrary geodesic balls A and B in (R n , F ε ), then (R n , F ε ) is a (locally) Minkowski space, i.e., w x is independent of x.
Minkowski spaces are the simplest non-Euclidean Finsler structures. However, it turns out that the equality in the Brunn-Minkowski inequality on a generic Minkowski space (R n , F ) is not automatically verified even for forward and backward geodesic balls, i.e., B + (x, r) = {y ∈ R n : F (y − x) < r} and B − (x, r) = {y ∈ R n : F (x − y) < r}. In addition, in Example 4.2 we provide two classes of Minkowski spaces where equalities in the Brunn-Minkowski inequality generate two genuinely different scenarios. (ii) B − (y − x 0 , R) = B + ( R r x, R) for some x 0 ∈ R. Proof. Any Minkowski space is a forward/backward complete Berwald space with zero flag curvature; thus Corollary 4.2 applies, yielding the validity of (4.30).
Assume that for the convex sets A and B we have equality in (4.30). Note that the geodesics in (R n , F ) are straight lines and for every x, y ∈ R n the Finslerian distance is given by d F (x, y) = F (y−x). Thus, the positive homogeneity of F implies that Z s (A, B) = (1 − s)A + sB. We also recall that m F (S) = L n (S) for every measurable set S ⊂ M. Accordingly, the equality in (4.30) can be transposed to an equality in the Euclidean Brunn-Minkowski inequality for A and B, obtaining that A and B are homothetic.
In the sequel, let A = B + (x, r) and B = B − (y, R) for some x, y ∈ R n and r, R > 0. (i)⇒(ii). Assume we have equality in (4.30) for A and B. Note that these sets are strictly convex domains of R n in the usual sense, both of them inheriting the convexity of the Minkowski norm F , see e.g. Bao, Chern and Shen [4, p. 12]. Accordingly, from the first part of the proof, the sets A and B are homothetic, i.e., B − (y, R) = c 0 B + (x, r) + x 0 , for some c 0 > 0 and x 0 ∈ R n . Moreover, it follows that c 0 = R r , thus B − (y − x 0 , R) = B + ( R r x, R). For simplicity, in the following example we restrict our argument to two-dimensional objects. where Q is a 2×2 positive definite symmetric matrix, ·, · is the usual scalar product in R 2 and b ∈ R 2 is fixed such that Q −1 b, b < 1; here Q −1 denotes the inverse of Q. The pair (R 2 , F b ) is a Randerstype Minkowski plane which describes the anisotropic Luneburg-type refraction in optical crystals or the electromagnetic field of the physical space-time in general relativity (in higher dimension), see Randers [34]. Note that (R 2 , F b ) is reversible if and only if b = (0, 0). Let R, r > 0 and x, y ∈ R 2 be arbitrarily fixed. Since the forward and backward indicatrices I + (x, r) = ∂B + (x, r) and I − (y, R) = ∂B − (y, R) are ellipses which can be obtained from each other by translation and dilation, equality in the Brunn-Minkowski inequality (4.30) holds for any choice of A = B + (x, r) and B = B − (y, R) in (R 2 , F b ), due to Corollary 4.3; see also Figure 1 where α ∈ [0, π/2), v > 0 and g ≈ 9.81. If we assume that g sin α < v, it turns out that (R 2 , F α ) is a Minkowski plane, describing the law of walking with a constant speed v[m/s] under the effect of gravity on a mountain slope having the angle α w.r.t. the horizontal plane, see Matsumoto [27]. It is clear that (R 2 , F α ) is reversible if and only if α = 0, which corresponds to the Euclidean setting and F b reduces to the standard (reversible) metric F 0 (y 1 , y 2 ) = y 2 1 + y 2 2 /v. Let x, y ∈ R n and r, R > 0 be arbitrarily fixed. We notice that the indicatrices I + (x, r) = ∂A = {z ∈ R 2 : F α (z − x) = r} and I − (y, R) = ∂B = {z ∈ R 2 : F α (y − z) = R} are convex limaçons which cannot be obtained from each other by dilation and translation, unless α = 0 (i.e., the mountain slope vanishes), see also Figure 1(b). Thus, due to Corollary 4.3, for α = 0 (i.e., we are in the non-Euclidean setting) any choice of A = B + (x, r) and B = B − (y, R) in (R 2 , F α ) provides strict inequality in the Brunn-Minkowski inequality (4.30).