Polya-Szego inequality and Dirichlet $p$-spectral gap for non-smooth spaces with Ricci curvature bounded below

We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called $CD(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{1,p}$-Sobolev norm decreases under such a rearrangement and apply the result to show sharp spectral gap for the $p$-Laplace operator with Dirichlet boundary conditions (on open subsets), for every $p\in (1,\infty)$. This extends to the non-smooth setting a classical result of B\'erard-Meyer and Matei; remarkable examples of spaces fitting out framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci$\geq K>0$, finite dimensional Alexandrov spaces with curvature$\geq K>0$, Finsler manifolds with Ricci$\geq K>0$. In the second part of the paper we prove new rigidity and almost rigidity results attached to the aforementioned inequalities, in the framework of $RCD(K,N)$ spaces, which seem original even for smooth Riemannian manifolds with Ricci$\geq K>0$.


Introduction
In 1884 Lord Rayleigh, in his book about the theory of sound [Ray], conjectured that, among all membranes of a given area, the disk has the lowest fundamental frequency of vibration. This was proven in 1920ies by Faber [Fa23] and Krahn [Kr25] for domains in the Euclidean plane and extended by Krahn [Kr26] to higher dimensions. The celebrated Rayleigh-Faber-Krahn inequality reads as follows. (1.1)

Polya-Szego and p-spectral gap in CD(K, N) spaces
In order to discuss the main results of the paper let us introduce some preliminaries about nonsmooth spaces with Ricci curvature bounded below in a synthetic sense. A metric measure space (m.m.s. for short) is a triple (X, d, m) where (X, d) is a compact metric space endowed with a Borel probability measure m with supp(m) = X, playing the role of reference volume measure. Using optimal-transport techniques, Lott-Villani [LV09] and Sturm [St06a,St06b] introduced the so called curvature-dimension condition CD(K, N ): the rough geometric picture is that a m.m.s. satisfying CD(K, N ) should be thought of as a possibly non-smooth metric measure space with Ricci curvature bounded below by K ∈ R and dimension bounded above by N ∈ (1, ∞) in a synthetic sense. The basic idea of this synthetic point of view is to consider weighted convexity properties of suitable entropy functionals along geodesics in the space of probability measures endowed with the quadratic transportation distance. A first technical assumption throughout the paper is the so called essentially non-branching property [RS14], which roughly amounts to require that the L 2 -optimal transport between two absolutely continuous (with respect to the reference measure m) probability measures moves along a family of geodesics with no intersections, i.e. a non-branching set of geodesics (for the precise definitions see Section 2.1). The class of essentially non-branching CD(K, N ) spaces is very natural for extending the Polya-Szego/Bérard-Meyer results. Indeed a key ingredient for both is the isoperimetric inequality (via a coarea formula argument) and it was proved by Cavalletti with the first author [CM17a] that the Lévy-Gromov isoperimetric inequality extends to essentially non-branching CD(K, N ) spaces (see Section 2.2 for the details). Examples of essentially non-branching CD(K, N ) spaces are Riemannian manifolds with Ricci curvature bounded below, finite dimensional Alexandrov spaces with curvature bounded below, Ricci limits and more generally RCD(K, N )-spaces, Finsler manifolds endowed with a strongly convex norm and with Ricci bounded below; let us stress that our results are new in all these celebrated classes of spaces (apart from smooth manifolds). A standard example of a space failing to satisfy the essential non-branching property is R 2 endowed with the L ∞ norm.
In order to state the main theorems, let us introduce some notation about the model onedimensional space and the corresponding monotone rearrangement. For any K > 0 and 1 < N < +∞ we define the one dimensional model space (I K,N , d eu , m K,N ) for the curvature dimension condition of parameters K and N by (1.3) It is not difficult to check that the distribution function µ is non increasing and left-continuous. We will let u # be the generalized inverse of µ, defined in the following way: For an arbitrary Borel function u : Ω → (−∞ + ∞), let u * be the monotone rearrangement of |u|. Finally, we denote by W 1,p 0 (Ω) the closure (with respect to the W 1,p -topology) of the set of Lipschitz functions compactly supported in Ω (see Section 2 for more details). We can now state the first main result of the paper. (1.5) Theorem 1.4 will be proved in Section 3. The two main ingredients in the proof are the coarea formula and the Lévy-Gromov isoperimetric inequality, though the full argument requires some work and several intermediate results.
The second main result is a spectral gap for the p-Laplacian with Dirichlet boundary conditions, in the spirit of Berard-Meyer-Matei Theorem 1.2. In order to state it we need to introduce some more notation. For every v ∈ (0, 1), let r(v) ∈ I K,N be such that v = m K,N ([0, r(v)]). For any fixed 1 < p < +∞, for any v ∈ (0, 1) and for any choice of K > 0 and 1 < N < +∞, define For any metric measure space (X, d, m) with m(X) = 1, for any open subset Ω ⊂ X and for any 1 < p < +∞, define : u ∈ LIP c (Ω; [0, +∞)) and u ≡ 0 .
Observe that for any 2 ≤ N ∈ N and K > 0, The spectral gap in CD(K, N ) spaces for Neumann boundary conditions, called Lichnerowicz inequality, was established by Lott-Villani [LV07] in case p = 2 (see also [EKS15] and [JZ16] for related results in RCD(K, N ) spaces) and by Cavalletti with the first author [CM17b] for general p ∈ (1, ∞) with different techniques.

Rigidity and almost rigidity in RCD(K, N) spaces
In order to discuss the rigidity statements associated to Theorem 1.4 and Theorem 1.5 let us recall the "Riemannian" refinement of the CD condition, called RCD. Introduced by Ambrosio-Gigli-Savaré [AGS14b] in case N = ∞ (see also [AGMR15]), the RCD condition is a strengthening of the CD condition by the requirement that the Sobolev space W 1,2 ((X, d, m)) is Hilbert (or, equivalently, the heat flow, or equivalently the laplacian, is linear). The main motivation is that the CD condition allows Finsler structures while the RCD condition isolates the "Riemannian" spaces.
A key property of the RCD condition is that, as well as CD, is stable under measured Gromov-Haudorff convergence [AGS14b,GMS15]. The finite dimensional refinement was subsequently proposed and throughly studied in [G15a, EKS15, AMS15] (see also [CaMi16]). We refer to these papers and references therein for a general account on the synthetic formulation of the latter Riemannian-type Ricci curvature lower bounds; for a survey of results, see the Bourbaki seminar [V18] and the recent ICM-Proceeding [Am18].
We can now state the rigidity result associated to the Polya-Szego inequality Theorem 1.4. In order to simplify the notation we will consider K = N − 1, the case of a general K > 0 follows by a scaling argument (recall that (X, d, m) is an RCD(K, N ) space for some K > 0 and 1 < N < +∞ if and only if the rescaled space ( sin Y . If moreover the function u achieving equality in the Polya-Szego inequality (3.19) is Lipschitz and |∇u| (x) = 0 for m-a.e. x ∈ supp(u), then u is radial; i.e. u = f (d(·, x 0 )), where x 0 is a tip of a spherical suspension structure of X and f : [0, π] → R satisfies |f | = u * .
When specialized to the smooth setting, the last result reads as follows.
Corollary 1.7 (Rigidity in the Polya-Szego inequality-Smooth Setting). Let (M, g) be an Ndimensional Riemannian manifold, N ≥ 2, with Ric g ≥ (N − 1)g and denote by m the normalized Riemannian volume measure. Let Ω ⊂ X be an open subset with m(Ω) ∈ (0, 1). Assume that for some p ∈ (1, ∞) there exists u ∈ W 1,p 0 (Ω), u ≡ 0, achieving equality in the Polya-Szego inequality (1.5). Then (M, g) is isometric to the round sphere S N of constant sectional curvature one. If moreover the function u achieving equality in the Polya-Szego inequality (3.19) is Lipschitz and Let us mention that our proof of both Theorem 1.6 and Corollary 1.7 builds on top of the almost rigidity in Lévy-Gromov inequality [CM17a] and seems new even in the smooth setting. The rough idea is that if the space X is not a spherical suspension then by the almost rigidity in Lévy-Gromov inequality, there is a gap in the isoperimetric profile of X and the model isoperimetric profile I N −1,N . Thus it is not possible to achieve almost equality in the Polya-Szego inequality for suitable approximations u n ∈ LIP c (Ω) of u with |∇u n | (x) = 0 m-a.e. x ∈ supp(u n ), hence contradicting that u ∈ W 1,p 0 (Ω) achieves equality in Polya-Szego inequality. The rigidity statement in the function is more subtle and basically consists in proving that the structure of spherical suspension induced by the optimality in Lévy-Gromov by every super-level set {u > t} is independent of t. Remark 1.8. A natural question about Theorem 1.6 regards sharpness of the assumptions. Clearly, if u ≡ 0 also the decreasing rearrangement u * vanishes; hence u, u * achieve equality in the Polya-Szego inequality but one cannot expect to infer anything on the space.
Let us also stress that the condition |∇u| = 0 m-a.e. is necessary to infer that u(·) = u * •d(x 0 , ·), even knowing a priori that the space is a spherical suspension with pole x 0 and that u achieves equality in Polya-Szego inequality. Indeed let X = S N be the round sphere, fix points x 1 = x 2 ∈ S N and radii 0 < r 1 < r 2 < r 3 such that B r1 (x 1 ) ⊂ B r2 (x 2 ) ⊂ B r3 (x 2 ). Consider a function u : S N → [0, 1] which is radially decreasing on B r1 (x 1 ) (with respect to the pole x 1 ), constant on B r2 (x 2 ) \ B r1 (x 1 ) and radially decreasing on B r3 (x 2 ) \ B r2 (x 2 ) (w.r.t. the pole x 2 ). It is easy to check that such a function u achieves equality in the Polya-Szego inequality but is not globally radial.
Our second rigidity result concerns the Dirichlet p-spectral gap. Theorem 1.9 (Rigidity for the Dirichlet p-spectral gap). Let (X, d, m) be an RCD(N −1, N ) space.
Let Ω ⊂ X be an open subset with m(Ω) = v for some v ∈ (0, 1) and suppose that λ p X (Ω) = λ p N −1,N,v . Then 2. the topological closureΩ of Ω ⊂ X coincides with the closed metric ball centred at one of the tips of the spherical suspension: 3. the eigenfunction u ∈ W 1,p 0 (Ω) associated to λ p X (Ω) is unique up to a scalar factor and it coincides with the radial one: i.e. called x 0 the centre of Ω and w : The proof of Theorem 1.9 builds on top of the rigidity in the Lévy-Gromov inequality proved in [CM17a]; indeed the rough idea to establish the first and second assertions is to prove that if λ p X (Ω) = λ p N −1,N,v , then the super-level sets of the first p-eigenfunction are optimal in the Lévy-Gromov inequality. The proof of the third assertion requires more work. The rough idea is to show that the first Dirichlet p-eigenfunction is unique thus, knowing already that Ω is almost a ball centred at a tip of the spherical suspension and hence there is already a natural radial first Dirichlet p-eigenfunction suggested by the model space, it follows that u must be radial. In the proof of the uniqueness of the first Dirichlet p-eigenfunction we have been inspired by a paper of Kawhol-Lindqvist [KL06] dealing with smooth Riemannian manifolds and, in order to implement the arguments in non-smooth setting, we make use of the theory of tangent modules of m.m.s. developed by Gigli [G18] (after Weaver [W99]).
Let us also mention that the rigidity for the Neumann spectral gap, known as Obata Theorem, was established in case p = 2 by Ketterer [K15] and by Cavalletti with the first author [CM17b] for general p ∈ (1, ∞).
We conclude the introduction with an almost-rigidity result which seems interesting even in the smooth framework, i.e. if (X, d, m) is an N -dimensional Riemannian manifold with Ricci curvature bounded below by N − 1. Let us point out that in [Be05] some related almost rigidity results have been obtained in the smooth setting for p = 2 under the additional assumption that the domain Ω is (mean) convex and m(Ω) ≤ 1/2. A key point in the proof is that the class of RCD(N − 1, N ) spaces is compact with respect to mGH convergence, a fact which clearly fails in the smooth setting as the limits usually present singularities. We denote by d mGH the measured Gromov Hausdorff distance between two normalized compact metric measure spaces. Then there exists a spherical suspension

Acknowledgement:
The work was developed while A.M. was based in the Mathematics Institute at the University of Warwick and, partly, when D.S. was visiting the institute. They would like to thank the institute for the excellent working conditions and stimulating environment. A.M. is supported by the EPSRC First Grant EP/R004730/1 "Optimal transport and geometric analysis" and by the ERC Starting Grant 802689 "CURVATURE". The authors wish to thank L. Ambrosio for inspiring discussions around the topics of the paper.

Preliminaries
Throughout the paper (X, d, m) will be a complete and separable metric measure space with supp(m) = X and m(X) < ∞. We will denote by B(X) the family of Borel subsets of X and by LIP(X) the space of real valued Lipschitz functions over X. For any open domain Ω ⊂ X, LIP c (Ω) and LIP loc (Ω) will stand for the space of Lipschitz functions with compact support in Ω and the space of locally Lipschitz functions in Ω. Given u ∈ LIP loc (X), its slope |∇u| (x) is defined as if x is not isolated 0 otherwise, moreover we introduce the notation Lip(u) for the global Lipschitz constant of u ∈ LIP(X). For any interval I ⊂ R we will denote by AC(I; X) the space of absolutely continuous curves γ : I → X. For any γ ∈ AC(I; X), the metric derivative |γ ′ | : I → [0, +∞] defined by provides the following representation of the the length of γ: Next we introduce Sobolev functions and Sobolev spaces over (X, d, m). We refer for instance to [AGS14a,ACDM15] for a detailed discussion about this topic. Definition 2.1 (Sobolev spaces and p-energy). Fix any 1 < p < +∞. The p-Cheeger energy Ch p : L p (X, m) → [0, +∞] is a convex L p (X, m)-lower semicontinuous functional defined by Moreover, we define W 1,p (X, d, m) := { Ch p < +∞ } and we remark that, when endowed with the norm By looking at the optimal approximating sequence in (2.1) one can find a minimal object called minimal weak upper gradient |∇f | w , providing the integral representation for any f ∈ W 1,p (X, d, m). We remark that without further regularity assumptions on the metric measure space the minimal weak upper gradient depends also on the integrability exponent p; nevertheless we will always omit this dependence in the notation.

Definition 2.2 (Local Sobolev spaces)
. Given an open set Ω ⊂ X, for any 1 < p < +∞ we will denote by W 1,p 0 (Ω) the closure of LIP c (Ω) in W 1,p (X, d, m), with respect to the W 1,p norm. We denote with P(X) the space of all Borel probability measures over X and with P 2 (X) the space of probability measures with finite second moment. The space P 2 (X) can be endowed with the L 2 -Kantorovich-Wasserstein distance W 2 defined as follows: for µ 0 , µ 1 ∈ P 2 (X), set where the infimum is taken over all π ∈ P(X × X) with µ 0 and µ 1 as the first and the second marginal. The space (X, d) is geodesic if and only if the space (P 2 (X), W 2 ) is geodesic.
A set G ⊂ Geo(X) is a set of non-branching geodesics if and only if for any γ 1 , γ 2 ∈ G, it holds: In the paper we will mostly consider essentially non-branching spaces, let us recall their definition (introduced by T. Rajala and Sturm [RS14]).

Definition 2.3.
A metric measure space (X, d, m) is essentially non-branching (e.n.b. for short) if and only if for any µ 0 , µ 1 ∈ P 2 (X), with µ 0 , µ 1 absolutely continuous with respect to m, any element of OptGeo(µ 0 , µ 1 ) is concentrated on a set of non-branching geodesics.
It is clear that if (X, d) is a smooth Riemannian manifold then any subset G ⊂ Geo(X) is a set of non branching geodesics, in particular any smooth Riemannian manifold is essentially non-branching.
In order to formulate curvature properties for (X, d, m) we recall the definition of the distortion coefficients: where the σ-coefficients are defined as follows: given two numbers K, N ∈ R with N ≥ 0, we set if Kθ 2 ≤ 0 and N > 0. (2.4) Let us also recall the definition of the Rényi Entropy functional E N : P(X) → [0, ∞]: The curvature-dimension condition was introduced independently by Lott-Villani [LV09] and Sturm [St06a,St06b], let us recall its definition.
Definition 2.4 (CD condition). Let K ∈ R and N ∈ [1, ∞). A metric measure space (X, d, m) verifies CD(K, N ) if for any two µ 0 , µ 1 ∈ P 2 (X, d, m) with bounded support there exist ν ∈ OptGeo(µ 0 , µ 1 ) and π ∈ P(X × X) W 2 -optimal plan such that µ t := (e t ) ♯ ν ≪ m and for any ( In particular if N = n the generalized Ricci tensor Ric g,h,N = Ric g makes sense only if h is constant. The lack of the local-to-global property of the CD(K, N ) condition (for K/N = 0) led in 2010 Bacher and Sturm to introduce in [BS10] the reduced curvature-dimension condition, denoted by CD * (K, N ). The CD * (K, N ) condition asks for the same inequality (2.6) of CD(K, N ) to hold but the coefficients τ (s) K,N (d(γ 0 , γ 1 )) are replaced by the slightly smaller σ (s) K,N (d(γ 0 , γ 1 )).
Since the CD condition allows Finsler geometries, in order to single out the "Riemannian" structures Ambrosio-Gigli-Savaré [AGS14b] introduced the Riemannian curvature dimension condition RCD(K, ∞) (see also [AGMR15] for the extension to σ-finite measures and for the present simplification in the axiomatization). A finite dimensional refinement, coupling the CD(K, N ) condition for N < +∞ with infinitesimal Hilbertianity, has been subsequently proposed in [G15a] while the the refined RCD * (K, N ) condition has been introduced and extensively investigated in [EKS15,AMS15]. We refer to these papers and references therein for a general account on the synthetic formulation of the latter Riemannian-type Ricci curvature lower bounds; for a survey of results, see the Bourbaki seminar [V18] and the recent ICM-Proceeding [Am18]. Here we only briefly recall that it is a stable [AGS14b,GMS15] strengthening of the reduced curvature-dimension condition: a m.m.s. verifies RCD * (K, N ) if and only if it satisfies CD * (K, N ) and is infinitesimally Hilbertian, meaning that the Sobolev space W 1,2 (X, m) is a Hilbert space (with the Hilbert structure induced by the Cheeger energy).
To conclude we recall also that recently Cavalletti and E. Milman [CaMi16] proved the equivalence of CD(K, N ) and CD * (K, N ), together with the local-to-global property for CD(K, N ), in the framework of essentially non-branching m.m.s. having m(X) < ∞. As we will always assume the aforementioned properties to be satisfied by our ambient m.m.s. (X, d, m), we will use both formulations with no distinction. It is worth also mentioning that a m.m.s. verifying RCD * (K, N ) is essentially non-branching (see [RS14, Corollary 1.2]) implying also the equivalence of RCD * (K, N ) and RCD(K, N ).
For all the main results we will assume that the m.m.s. (X, d, m) is essentially non-branching and satisfies CD(K, N ) from some K > 0 with supp(m) = X (or, more strongly, that (X, d, m) is a RCD(N − 1, N ) space). It follows that (X, d) is a geodesic and compact metric space with m(X) ∈ (0, ∞). Since (X, d, m) is a CD(K, N ) (resp. RCD(K, N )) space if and only if (X, d, αm) is so, without loss of generality we will also assume m(X) = 1.

Finite perimeter sets and Lévy-Gromov isoperimetric inequality
We now recall the definition of a finite perimeter set in a metric measure space (see [Am02,Mi03] and the more recent [ADM14]).

Definition 2.5 (Perimeter and sets of finite perimeter). Given a Borel set E ⊂ X and an open set
A the perimeter Per(E, A) is defined in the following way: We say that E has finite perimeter if Per(E, X) < +∞; in this case, we shall denote Per(E) := Per(E, X) to simplify the notation. It can be proved that the set function A → Per(E, A) is the restriction to open sets of a finite Borel measure Per(E, ·) defined by Below we recall the definition of the family of one dimensional model spaces for the curvature dimension condition of parameters K > 0 and 1 < N < +∞ (cf. [Gr07, Appendix C] and [M15]).
Definition 2.6 (One dimensional model spaces). For any K > 0 and for any 1 < N < +∞ we define the one dimensional model space (I K,N , d eu , m K,N ) for the curvature dimension condition of parameters K and N by where d is the restriction to I K,N of the canonical Euclidean distance over the real line and c K,N := In order to shorten the notation, we set h K,N (t) := 1 cK,N sin t Let us recall that, for any normalized metric measure space (X, d, m), the isoperimetric profile We will denote by I K,N the isoperimetric profile of the model space ( In [CM17a,CM18], exploiting the so-called localization technique (cf. [Kl17]), the following version of the Lévy-Gromov isoperimetric inequality [Gr07, Appendix C] for metric measure spaces was proven.
Theorem 2.7 (Lévy-Gromov inequality). Let (X, d, m) be an essentially non branching CD(K, N ) metric measure space for some K > 0 and 1 < N < +∞. Then, for any Borel set E ⊂ X, it holds We next recall the notion of warped product between metric measure spaces, generalizing the well know Riemannian construction. Given two geodesic metric measure spaces (B, d B , m B ) and (F, d F , m F ) and a Lipschitz function f : B → [0, +∞) one can define a length structure on the product B × F as follows: for any absolutely continuous curve γ : and consider the associated pseudo-distance The f -warped product of B with F is the metric space defined by where (p, x) ∼ (q, y) if and only if d((p, x), (q, y)) = 0. One can also associate a natural measure and obtain In [CM17a,CM18] also the rigidity problem for the Lévy-Gromov inequality was addressed in the framework of metric measure spaces. Before stating the result from [CM17a,CM18], observe that if (X, d, m) is an RCD(K, N ) metric measure space for some K > 0 and 1 < N < +∞ then Moreover in this case the following hold:

BV functions and coarea formula in m.m.s.
As for the classical Euclidean case, in metric measure spaces one can introduce not only a notion of finite perimeter set but also a notion of function of bounded variation. We refer again to [Mi03] and [ADM14] for more details about the topic.
By localizing this construction one can define for any open A ⊂ X. In [ADM14] it is proven that this set function is the restriction to open sets of a finite Borel measure that we call total variation of f .
For any Lipschitz function f : X → R it is easy to check that f ∈ BV * (X, d, m) and |Df | * ≤ |∇f | m. In the following we will denote by |∇f | 1 the density of |Df | * with respect to m. With a slight abuse of notation motivated by simplicity, we are going to use the same symbol |∇f | 1 to denote the equivalence class (under m-a.e. equality) and a Borel representative. The following result is a simplified version of [APS15, Proposition 4.2]. Proposition 2.10. Let f ∈ LIP(X). Then |∇f | 1 (x) = 0 for m-a.e. x ∈ {f = 0}.
Let us point out that on general metric measure spaces there is no reason to expect any identification, in the almost everywhere sense, between |∇f | 1 , |∇f | and the p-minimal weak upper gradient |∇f | w , for some p > 1, of a Lipschitz function f . The identification result stated below is a consequence of the seminal work [Ch99] concerning Lipschitz functions on metric measure spaces satisfying doubling and Poincaré inequalities and of the identification result for p-minimal weak upper gradients obtained in [GH14] for proper RCD(K, ∞) spaces. In particular in [GH14, Remark 3.5] it is observed that, for any Lipschitz function f , the density of |Df | * with respect to m coincides with the p-minimal weak upper gradient for any p > 1. In [Ch99] instead it is proved that the slope coincides with the p-minimal weak upper gradient m-almost everywhere. Combining these two ingredients we get: Then for any f ∈ LIP(X) one has that |Df | * = |∇f | m.
The following coarea formula for functions of bounded variation on metric measure spaces is taken from [Mi03,Remark 4.3]. It will play a key role in the rest of the paper. (2.8) Combining Proposition 2.11 and Theorem 2.12 we obtain the following. (2.9) The following result will be useful when dealing with the almost rigidity case in the spectral gap inequality. We refer to [V09,Chapter 27] (see also [GMS15, Section 3.5]) for the relevant background about measured Gromov-Hausdorff convergence.
Proposition 2.14. Fix K > 0 and N ∈ (1, ∞). Let ((X n , d n , m n )) n∈N be a sequence of normalized RCD(K, N ) spaces converging to (X, d, m) in the measured Gromov-Hausdorff sense. Denote by I n (resp. I) the isoperimetric profile of (X n , d n , m n ) (resp. of (X, d, m)). Then, for any t ∈ [0, 1] and for any sequence (t n ) n∈N with t n → t, it holds that (2.10) Proof. We refer to [GMS15,AH16] for the basic definitions and statements about convergence of functions defined over mGH-converging sequences of metric measure spaces.
First of all note that in order to prove (2.10), without loss of generality we can assume that sup n∈N I n (t n ) < +∞. For any n ∈ N let E n ⊂ X n be a Borel set such that Per n (E n ) = I n (t n ), whose existence follows as in the Euclidean case from standard lower semicontinuity and compactness arguments. We now claim that f is the indicator function of a Borel set E ⊂ X, with m(E) = t. To this aim call g n := χ En (1 − χ En ) and observe that (g n ) n∈N strongly L 1 -converges to g := f (1 − f ) thanks to [AH16, Proposition 3.3]. Thus g = 0, since g n = 0 for any n ∈ N and therefore g is the indicator function of a Borel set, as claimed.
We can now apply [AH16, Theorem 8.1] to get the Mosco convergence of the BV energies and conclude that The lower semicontinuity for the isoperimetric profiles (2.10) easily follows, since E is an admissible competitor in the definition of I(t).

Polya-Szego inequality
The working assumption of this section, unless otherwise stated, is that (X, d, m) is an essentially non branching CD(K, N ) space for some K > 0, N ∈ (1, +∞), with m(X) = 1 and supp(m) = X. It is not difficult to check that the distribution function µ is non increasing and left-continuous. If moreover u is continuous, then µ is strictly decreasing. We let u # be the generalized inverse of µ, defined in the following way: x])), ∀x ∈ [0, r]. For simplicity of notation we will often write u * in place of u * K,N . Remark 3.4. For simplicity of notation, throughout the paper we will consider monotone rearrangements of non-negative functions. Nevertheless, for an arbitrary Borel function u : Ω → (−∞ + ∞) the analogous statements hold by setting u * the monotone rearrangement of |u|.
In the next proposition we collect some useful properties of the monotone rearrangement, whose proof in the Euclidean setting can be found for instance in [K06, Chapter 1] and can be adapted with minor modifications to our framework.  Then for any non-negative u ∈ LIP c (Ω) with |∇u| p dm < ∞, there exists a sequence (u n ) n∈N with u n ∈ LIP c (Ω) non-negative, |∇u n | = 0 m-a.e. on { u n > 0 } for any n ∈ N and such that u n → u in W 1,p (X, m). In particular |∇u n | p dm → |∇u| p dm as n → ∞.
Proof. It is straightforward to check that there exists a sequence (ǫ n ) n∈N monotonically converging to 0 from above such that m({|∇u| = ǫ n }) = 0 for any n ∈ N.
Choose an open set Ω ′ containing the support of u and compactly contained in Ω. Let v : Ω → [0, +∞) be the distance function from the complementary of Ω ′ in X, namely v(x) := dist(x, X \ Ω ′ ) for any x ∈ Ω.
Observe that v ∈ LIP c (Ω), moreover Indeed it suffices to observe that the restriction of v to any geodesic connecting x with y ∈ X \ Ω ′ such that v(x) = d(x, y) has slope equal to 1 at x. Next we introduce the approximating sequence u n := u + ǫ n v and we claim that it has the desired properties. Indeed, if u ∈ LIP c (Ω) is non-negative, then also u n ∈ LIP c (Ω) is so.  Fix any 1 < p < +∞ and let u, u n ∈ LIP c (Ω) be as in the statement and the proof of Lemma 3.6. Then, for any n ∈ N, it holds that |∇u n | 1 (x) = 0 for m-a.e. x ∈ { u n > 0 }.
Proof. One of the properties of the approximating sequence in Lemma 3.6 is that |∇u n | (x) = 0 for m-a.e. x ∈ { u n > 0 }. The desired conclusion follows from [APS15] where it is proved that, under the locally doubling and Poincaré assumption, there exists c > 0 such that for any function f ∈ LIP loc (X).
Remark 3.8. Since any essentially non branching CD(K, N ) metric measure space is (locally) doubling and verifies a weak 1-1 Poincaré inequality (see [VR08]), Corollary 3.7 applies to the case of our interest.
In Proposition 3.9 below we extend to the non smooth setting [K06, Theorem 2.3.2]; the key idea is to replace the Euclidean isoperimetric inequality with the Lévy-Gromov isoperimetric inequality Theorem 2.7. Proof. Let µ be the distribution function associated to u and denote by M := sup u < +∞. Observe that our assumptions grant continuity and strict monotonicity of µ. Therefore for any s, k ≥ 0 such that s + k ≤ m(Ω) we can find 0 ≤ t − h ≤ t ≤ M in such a way that µ(t − h) = s + k and µ(t) = s. Taking into account the assumption that u is L-Lipschitz we can say that (3.4) On the other hand, an application of the coarea formula (2.8) yields Applying the Lévy-Gromov isoperimetric inequality Theorem 2.7 we can estimate the right hand side of (3.5) in the following way: Recalling that the model isoperimetric profile I K,N is continuous and that µ is continuous, combining (3.4) with (3.6) and eventually applying the mean value theorem we get Calling u # the inverse of the distribution function, the estimate (3.7) can be rewritten as we obtain that 1 = d dt r(t)h K,N (r(t)) and, since we know that I K,N (s) = h K,N (r(s)), it follows that d dt . (3.10) By definition of the rearrangement u * , for any x ∈ I K,N it holds that u * (x) = u # (m K,N ([0, x])).
Combining the last identity with (3.9) and (3.10) we can estimate for x ≤ y The next lemma should be compared with [K06], dealing with the case of smooth functions in Euclidean domains. where the quantity 1/ |∇u| 1 is defined to be 0 whenever |∇u| 1 = 0.
Proof. Fix any ǫ > 0 and define Fixing t ≥ 0 and h > 0, an application of the coarea formula (2.8) with f = f ǫ yields to Now we pass to the limit as ǫ → 0 both at the right hand side and at the left hand side in (3.12).
The assumption that |∇u| 1 = 0 m-a.e. grants that the integrand at the left hand side monotonically converges m-a.e. to 1. Thus an application of the monotone convergence theorem yields that With the above mentioned convention about the value of 1/ |∇u| 1 at points where |∇u| 1 = 0, applying the monotone convergence theorem twice at the right hand side of (3.12), we get as ǫ goes to 0. Combining (3.12), (3.13) and (3.14), we get It follows that the distribution function is absolutely continuous and therefore differentiable at almost all points with derivative given by the explicit expression (3.11).
Before proceeding to the statement and the proof of the Polya-Szego inequality we need an identification result between slopes and 1-minimal weak upper gradients in the simplified setting of the model weighted interval I K,N . In this setting, for any p ≥ 1, we say that u ∈ W 1,p (I, d, m K,N ) if the distributional derivative of u (defined through integration by parts) is in L p (I, m K,N ). We refer to [ADM14,Section 8] for an account about W 1,1 spaces on metric measure spaces.
Lemma 3.11. Let I ⊂ I K,N be a sub-interval and let 1 < p < +∞. Let f ∈ W 1,p ((I, d, m K,N )) be monotone. Then f ∈ W 1,1 ((I, d, m K,N )) and it holds (3.15) Proof. The fact that f ∈ W 1,1 ((I, d, m K,N )) follows directly by Hölder inequality, since m K,N (I) ≤ 1. Since m K,N = h K,N L 1 with h K,N locally bounded away from 0 out of the two end-points of I K,N , it follows that f is locally absolutely continuous in the interior of I K,N . In particular it is differentiable L 1 -a.e. and |∇f | (x) = |f ′ | (x) at every differentiability point x. We are thus left to show the first equality in (3.15). Note that the assumptions ensure that f is invertible onto its image, up to a countable subset of f (I). The coarea formula in the 1-dimensional case reads as On the other hand, the change of variable formula via a monotone absolutely continuous function gives The combination of (3.16) with (3.17) then gives the first equality in (3.15).
The following statement should be compared with [K06], where the study of the monotone rearrangement on domains of R n is performed. respectively. An application of Hölder's inequality yields that for any where µ denotes the distribution function associated to u. It follows from the discussion above and from Lemma 3.10 that L 1 -a.e. point t ∈ (0, M ) is a differentiability point of both µ, φ and ψ. In view of (3.20), at any such point it holds that Applying the Lévy-Gromov inequality Theorem 2.7 we obtain that Per({u > t}) ≥ I K,N (µ(t)). Therefore, taking into account the strict monotonicity of µ, (3.21) turns into It follows from the very definition of the monotone rearrangement and from the properties of the model isoperimetric profile that Per({u * > t}) = I K,N (µ(t)) (recall that u and u * have the same distribution function). Moreover, since we already know that u * is Lipschitz, we are in position to apply Lemma 3.10 to conclude (taking also into account Lemma 3.11) that (3.24) Applying the coarea formula to the function u * and taking into account (3.24) and Lemma 3.11 we conclude that Comparing (3.23) with (3.25) we can conclude that giving (3.18). To get (3.19) it suffices to recall that |∇u| 1 ≤ |∇u| holds true m-a.e..
Armed with Proposition 3.12 we can extend the celebrated Polya-Szego inequality to the nonsmooth CD(K, N ) framework.

Proof of Theorem 1.4.
By the very definition of W 1,p 0 (Ω) we can find a sequence (u n ) n∈N with u n ∈ LIP c (Ω) for any n ∈ N and u n converging to u in W 1,p (X, d, m), Moreover, thanks to Lemma 3.6, we can assume that |∇u n | 1 = 0 for m-a.e. x ∈ { u n > 0 } for any n ∈ N, so that we can apply Proposition 3.12 to each of the functions u n obtaining Hence, taking into account (3.26) and the strong convergence in W 1,p (X, d, m) of u n to u, we conclude that which is the desired conclusion.
In the following we will need an improved version of the Polya-Szego inequality. To this aim, for any non-negative u ∈ W 1,p 0 (Ω) we introduce a function f u : Observe that this definition makes sense thanks to Theorem 1.4 and the coarea formula, which also yields for any u ∈ W 1,p 0 (Ω). We are now in position to state and prove our improved Polya-Szego inequalities. Suppose that u ∈ W 1,p 0 (Ω) is such that u * has non vanishing derivative L 1 -a.e. on (0, r). Then As a consequence, under the same assumptions, it holds that Proof. In order to prove (3.29) we just need to observe that our assumptions, even though being weaker than those of Proposition 3.12, put us in position to make its proof work. Indeed, with the same notation therein introduced, we observe that the monotone rearrangement u * has the same distribution function of u. Moreover, Theorem 1.4 implies in particular that u * ∈ AC loc ((0, r)). Therefore, since we are assuming that |∇u * | (t) = 0 for L 1 -a.e. t, it follows from Lemma 3.10 (taking into account also Lemma 3.11) that µ is absolutely continuous and therefore differentiable L 1 -a.e. with the explicit expression for the derivative given (for L 1 -a.e. t) by , where the second equality is a consequence of the very construction of the monotone rearrangement.
Following verbatim the beginning of the proof of Proposition 3.12 we obtain that (3.22) is still valid in the present setting. Taking into account (3.31) we obtain that for L 1 -a.e. t ∈ (0, sup u * ). The desired inequality (3.29) follows now recalling that Conclusion (3.30) is a consequence of (3.29) after observing that {u > t} is an admissible competitor in the definition of I (X,d,m) (µ(t)) since, by definition, it holds that m({u > t}) = µ(t).
Remark 3.14. In order to prove the forthcoming Theorem 5.5 we will need to slightly enlarge the class of functions where (3.29) and (3.30) hold true.
In particular, we claim that (3.29) holds true for any u ∈ W 1,p 0 (Ω) such that u * is C 1 and strictly decreasing. Indeed for any such u it holds that the set of critical values of u * is L 1 -negligible. Moreover, the distribution function µ of u (which coincides with the distribution function of u * by equimeasurability, as we already observed), is differentiable at any regular point of u * , with derivative given by (3.31). Hence the whole proof of Proposition 3.13 can be carried over without modifications.

Bérard-Meyer for essentially non-branching CD(K, N) spaces
The goal of this section is to bound from below the p-spectral gap of an essentially non branching CD(K, N ) space with the one of the corresponding one dimensional model space, for any K > 0, N ∈ (1, +∞) and p ∈ (1, +∞). This extends to the non-smooth setting the celebrated result of Bérard-Meyer [BM82] (see also [Ma00]) proved for smooth Riemannian manifolds with Ric ≥ K, K > 0.
For every K > 0, N ∈ (1, +∞), let (I K,N , d eu , m K,N ) be the one dimensional model space defined in (2.7). For every v ∈ (0, 1), let r(v) ∈ I K,N be such that v = m K,N ([0, r(v)]). To let the notation be more compact, for any fixed 1 < p < +∞, for any v ∈ (0, 1) and for any choice of K > 0 and 1 < N < +∞, we define [0, +∞)), u(r(v)) = 0 and u ≡ 0 , and we call λ p K,N,v the comparison first eigenvalue for the p-Laplacian with Dirichlet boundary conditions for Ricci curvature bounded from below by K, dimension bounded from above by N and volume v.
Moreover, for any metric measure space (X, d, m) with m(X) = 1, for any open subset Ω ⊂ X and for any 1 < p < +∞, we define : u ∈ LIP c (Ω; [0, +∞)) and u ≡ 0 , (4.1) and we call λ p X (Ω) the first eigenvalue of the p-Laplacian on Ω with Dirichlet boundary conditions. Observe that for any 2 ≤ N ∈ N and K > 0, We are now in position to prove Theorem 1.5 Proof of Theorem 1.5 For any u ∈ LIP c (Ω; [0, +∞)) not identically zero we introduce the notation for the Rayleigh quotient of u. It follows from the combination of Proposition 3.5 and Proposition 3.12 that for any u ∈ LIP c (Ω; [0, +∞)) such that |∇u| 1 = 0 m-a.e. on { u > 0 } it holds where u * : [0, r(v)] → [0, +∞) is the monotone rearrangement of u on the model space. Observe now that u ∈ LIP c (Ω) implies, by construction of the monotone rearrangement, that u * vanishes at r(v). We thus get R p (u * ) ≥ λ p K,N,v . The desired conclusion follows from Corollary 3.7 and Remark 3.8 which grant that for any u ∈ LIP c (Ω; [0, +∞)) we can find a sequence (u n ) n∈N ⊂ LIP c (Ω; [0, +∞)) such that |∇u n | 1 = 0 m-a.e. on { u n > 0 } for any n ∈ N and R p (u n ) → R p (u), as n → ∞.

Existence of minimizers
Here we collect some known result about the p-Laplace equation with homogeneous Dirichlet boundary conditions on metric measure spaces (verifying the curvature dimension condition) that will be useful in the next section about rigidity. We refer to [LMP05] and [GM13] for a more detailed discussion about this topic and equivalent characterizations of first eigenfunctions.
Recall that we defined W 1,p 0 (Ω) to be the closure w.r.t. the W 1,p -norm of LIP c (Ω) (see Definition 2.2). In the fairly general context of metric measure spaces it makes sense to talk about the first eigenfunction of the p-Laplace equation if the notion is understood in the following weak sense.
Definition 4.1 (First eigenfunction). Let Ω ⊂ X be an open domain. We say that u ∈ W 1,p 0 (Ω) is a first eigenfunction of the p-Laplacian on Ω (with homogeneous Dirichlet boundary conditions) if u ≡ 0 and it minimizes the Rayleigh quotient among all functions v ∈ W 1,p 0 (Ω) such that v ≡ 0. Remark 4.2. Let us observe that if u ∈ W 1,p 0 (Ω) is a first eigenfunction of the p-Laplacian then R p (u) = λ p X (Ω) (that is the first eigenvalue of the p-Laplace equation defined in (4.1)), since by the very definition of the space W 1,p 0 (Ω) it makes no difference to minimize the Rayleigh quotient over LIP c (Ω) or over W 1,p 0 (Ω). As we will see below, the advantage of considering the minimization over W 1,p 0 (Ω) is to gain existence of minimizers. We conclude this short section with a general existence result for first eigenfunctions of the p-laplacian. The main ingredient for its proof, as in the smooth case, is the Sobolev inequality which implies in turn that also Rellich compactness theorem holds true in this setting. A good reference for this part is [AT04, Chapter 5]. Proof. If λ p X (Ω) < +∞, we can find a sequence (u n ) n∈N ⊂ W 1,p 0 (Ω) such that u n L p = 1 for any n ∈ N and |∇u n | w p L p → λ p X (Ω) as n → ∞. Recall that the CD(K, N ) condition for K > 0 and 1 < N < +∞ grants that X is a compact and doubling metric measure space. Hence we can apply [AT04,Theorem 5.4.3] (which is a general version of Rellich theorem for metric measure spaces) to the sequence (u n ) n∈N to find a limit function u ∈ W 1,p 0 (Ω) such that u n → u in L p (Ω, m) as n → ∞ and hence u L p = 1. It follows from the lower semicontinuity of the p-energy w.r.t. L p (Ω, m)-convergence that

Rigidity in the Polya-Szego inequality
This section is devoted to prove a rigidity statement associated to the Polya-Szego inequality Proposition 3.12. The rough idea here is that if equality occurs in the Polya-Szego inequality then it occurs in the Lévy-Gromov inequality too, and hence one can build on top of the rigidity statements in the Lévy-Gromov isoperimetric inequality established in [CM17a,CM18]. Let us also mention the paper [FV03], where an elementary proof of the rigidity statement for the Polya-Szego inequality on R n is presented.
, where x 0 is a tip of a spherical suspension structure of X.
Remark 5.2. Before discussing the proof, let us stress that Theorem 5.1 is stated for a non-negative function u just for uniformity of notation with the previous sections. Nevertheless, such a nonnegativity assumption can be suppressed, once the rearrangement u * in the Polya-Szego inequality (3.19) is understood as the decreasing rearrangement of |u| (see also Remark 3.4). The same holds for Theorem 5.4 below.

Proof.
Step 1: (X, d, m) is a spherical suspension. If equality occurs in (3.19), it follows from the proof of Proposition 3.12 that equality must occur in Step 2: for every t ∈ [0, M ), the closure of the open superlevel set {u > t} is a closed metric ball centred at a tip of a spherical suspension. We first claim that (5.1) holds for every t ∈ (0, M ). To this aim, call G ⊂ [0, M ] the subset of those t ∈ (0, M ) where (5.1) holds true. Since G is dense, for any fixed t ∈ [0, M ] we can find a sequence (t n ) n∈N ⊂ G such that t n → t as n → ∞. Our assumptions grant that {u > t n } converges in measure to {u > t}. From the lower semicontinuity of the perimeter and the continuity of the model isoperimetric profile it follows that: is an open set of m-measure zero, thus empty as supp m = X. Without loss of generality we can Let us stress that a priori the structure of spherical suspension may depend on t ∈ (0, M ); for instance in the N -sphere any point is a pole with respect to a corresponding structure of spherical suspension and any metric ball centred at any point is optimal for the isoperimetric problem.
Step 3: Conclusion. To prove the rigidity statement about the function u, we need to show that the above structure as spherical suspension is independent of t ∈ (0, M ). To this aim we first observe that, if equality holds in (3.19), then equality holds also in (3.21) for L 1 -a.e. t ∈ (0, M Since u * is Lipschitz with (u * ) ′ (t) < 0 for a.e. t ∈ (0, r(v)), it admits a strictly decreasing absolutely continuous inverse function that we denote by v * . We claim that f (x) := v * • u(x) is the distance function from a fixed point x 0 , playing the role of the pole of a structure as spherical suspension independent of t.
To this aim we first observe that the combination of (5.3) with (3.31) gives that |∇f | (x) = 1 for m-a.e. x ∈ { u > 0 }. From Step 2, we know that, for any t ∈ (0, max f ), {f < t} is a closed metric ball of radius r(t) centred at a point x t ∈ X; moreover X admits a structure of spherical suspension having x t as one of the two tips. In particular, for any t ∈ (0, max f ), it holds Combining what we obtained above with the assumption that |∇u| = 0 m-a.e. on supp u, we get that u has a unique maximum point for every t ∈ [0, max f ]. Therefore r ′ (t) = 1 for every t ∈ [0, max f ] and thus r(t) ≡ t.
We now claim that the centre x t of the ball B t (x t ) = { f < t } is independent of t.
If not we can find t ∈ (0, max f ) such that x t = x 0 . This implies that there exist ǫ > 0 and Since f (0) = 0 and f (x) = t, we claim that this contradicts |∇f | = 1 m-a.e. on supp u.
From the continuity of f , we can find δ ∈ (0, ǫ/8) such that for every x ′ ∈ B δ (x) and every and let (µ t ) t∈[0,1] be a W 2 -geodesic joining them. From [GRS16] the dynamic optimal transference plan ν representing (µ t ) t∈[0,1] is a test plan. We thus reach a contradiction: Remark 5.3. A natural question is whether the condition |∇u| = 0 m-a.e. is sharp in Theorem 5.1. Clearly, if u is a constant function, also the decreasing rearrangement u * is constant; hence u, u * achieve equality in the Polya-Szego inequality but one cannot expect to infer anything on the space. However in the next Theorem 5.4 we show that, as soon as u is non constant, the equality in Polya-Szego forces the space to be a spherical suspension. The proof of such a statement is more delicate than step 1 of Theorem 5.1 and builds on top of the almost rigidity for Lévy-Gromov inequality. As already observed in Remark 1.8, the condition |∇u| = 0 m-a.e. is necessary to infer that u(·) = u * • d(x 0 , ·), even knowing a priori that the space is a spherical suspension with pole x 0 and that u achieves equality in Polya-Szego.
Let Ω ⊂ X be an open set such that m(Ω) = v ∈ (0, 1) and assume that there exists a nonnegative function u ∈ W 1,p 0 (Ω), u ≡ 0, achieving equality in the Polya-Szego inequality (1.5). Then (X, d, m) is a spherical suspension, namely there exists an RCD(N − 2, Proof. Let (u n ) n∈N be a sequence of Lipschitz functions with compact support in Ω such that |∇u n | = 0 m-a.e. on { u n > 0 } for any n ∈ N approximating u in L p (Ω, m) and in W 1,p energy given by Lemma 3.6. Let u * n and u * be the decreasing rearrangements of u n and u respectively. The L p -continuity of the decreasing rearrangement, together with the lower semicontinuity of the p-energy and the Polya-Szego inequality, yield (5.5) It follows that (u * n ) n∈N converges in W 1,p -energy to u * , since by assumption u achieves the equality in the Polya-Szego inequality. Up to extracting a subsequence, that we do not relabel, we can assume that (u * n ) n∈N converges to u * both locally uniformly on (0, r(v)] and in W 1,p (([0, r(v)], d eu , m N −1,N )), and moreover that both the lim inf and the lim sup in (5.5) are full limits. Denoting by µ n and µ the distribution functions of u n and u respectively, it follows that, for any t ∈ (0, sup u * ) such that m N −1,N ({ u * = t }) = 0, it holds µ n (t) → µ(t) as n → ∞. To conclude we observe that, thanks to the assumption that u is non constant and to what we already observed, we can find 0 < t 0 < t 1 < t 2 < t 3 < sup u * , 0 < ǫ < 1 and n 0 ∈ N such that the following hold true: { t1<u * <t2 } |∇u * | p dm N −1,N > 0, (5.8) { t 1 < u * < t 2 } ⊂ { t 0 < u * n < t 3 } for any n ≥ n 0 (5.9) and µ n (t) ∈ [ǫ, 1 − ǫ] for any t ∈ [t 0 , t 3 ] and n ≥ n 0 . (5.10) Combining (5.9) with the L p (m N −1,N ) convergence of |∇u * n | to |∇u * | and the coarea formula, we obtain that (5.11) Eventually, putting (5.7) together with (5.8) and (5.11), we obtain lim inf

Rigidity in the p-spectral gap
The goal of this section is to investigate the rigidity in the p-spectral gap inequality (for Dirichlet boundary conditions), i.e. to prove Theorem 1.9.
Let Ω ⊂ X be an open set with m(Ω) = v for some v ∈ (0, 1) and suppose that λ p X (Ω) = λ p N −1,N,v . Then (X, d, m) is isomorphic to a spherical suspension and the topological closure of Ω coincides with the closed metric ball centred at one of the tips of the spherical suspension of m-measure v.
Proof. Suppose that λ p X (Ω) = λ p N −1,N,v . Let u ∈ W 1,p 0 (Ω) be a non-negative eigenfunction with u L p = 1 associated to the first eigenvalue λ p X (Ω), whose existence is granted by Theorem 4.3. Then Theorem 1.4 gives: where, as before, r is defined by m N −1,N ([0, r]) = m(Ω) = v. Hence equality holds true in all the inequalities so that u * is an eigenfunction of the p-Laplacian associated to the first eigenvalue on the one dimensional model space ([0, r], d eu , m N −1,N ). It follows from the corresponding ODE that u * ∈ C 0 ([0, r]) ∩ C 1 ((0, r)) and it is strictly decreasing. Hence, taking into account Remark 3.14, (3.29) holds true so that Therefore it must hold Per({u > t}) = I N −1,N (µ(t)), (5.12) for L 1 -a.e. t such that f u (t) = 0. In particular there exists at least one level t 0 such that the super-level set {u > t} is optimal for the Lévy-Gromov inequality. Thus we are in position to apply Theorem 2.8 to conclude that (X, d, m) is isomorphic, as a metric measure space, to a spherical suspension.
Moreover the C 1 regularity of u * , together with Sard's lemma, grants that the set of those levels t such that (5.12) holds true is dense in (0, sup u * ) (actually it is a full L 1 -measure set). In particular we can find a sequence (t n ) n∈N monotonically converging to 0 from above such that {u > t n } is optimal in the Lévy-Gromov inequality. It follows from the continuity of the model profile I N −1,N and the lower semicontinuity of the perimeter w.r.t. L 1 convergence that {u > 0} is optimal in the Lévy-Gromov inequality itself. Since {u > 0} is an open subset and supp(m) = X, part (iii) of Theorem 2.8 implies that there exists an RCD(N − 2, N − 1) space (Y, d Y , m Y ) such that X ≃ [0, π] × N −1 sin Y and is an open set of m-measure zero, hence it is empty (observe that {u > 0} has the same perimeter and the same measure of {u > 0}). Without loss of generality we can assume the first case holds and therefore [0, r) × Y ⊂ { u > 0 }. Note that the topological closure of [0, r) × Y is [0, r] × Y . Moreover, observing that [LMP05, Corollary 5.7] grants that u is strictly positive on Ω, we obtain that the topological closure of {u > 0} coincides withΩ, the topological closure of Ω. Applying once more part (iii) of Theorem 2.8 and taking into account the assumption that m has full support we can also say that { u > 0 } ⊂ [0, r]×Y . The desired conclusionΩ = [0, r] × Y follows.
Recall that, in the case of smooth Riemannian manifolds, the eigenfunction associated to the first eigenvalue on a smooth domain (with Dirichlet boundary conditions) is always simple (see for instance [KL06] for an elementary proof). In order to prove that in the case of rigidity in the spectral gap inequality also the eigenfunction must coincide with the radial one, we will exploit a generalization of such a principle to the case of our interest.
Theorem 5.6 (Eingenfunction-Rigidity for the p-spectral gap). Let (X, d, m) be an RCD(N −1, N ) space isomorphic to a spherical suspension and let Ω ⊂ X be an open subset whose topological closure coincides with a closed metric ball centred at one of the tips of the spherical suspension satisfying λ p X (Ω) = λ p N −1,N,v , m(Ω) = v ∈ (0, 1). Then, for any 1 < p < +∞, the eigenfunction associated to the first eigenvalue of the p-Laplace equation with homogeneous Dirichlet boundary conditions on Ω is unique up to a scalar factor (and it coincides with the radial one).
In order to conclude the proof we next show that v 2 ≡ cv 1 for some constant c ∈ R. To this aim, for any r 0 < r 1 ∈ (0, R) let Γ r0,r1 := {γ r0,r1 y,ε In view of what we remarked above, up to extracting a subsequence we can assume that (v n ) n∈N By the very definition of λ p