SHARP GEOMETRIC AND FUNCTIONAL INEQUALITIES IN METRIC MEASURE SPACES WITH LOWER RICCI CURVATURE BOUNDS

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1 SHAP GEOMETIC AND FUNCTIONAL INEQUALITIES IN METIC MEASUE SPACES WITH LOWE ICCI CUVATUE BOUNDS FABIO CAVALLETTI AND ANDEA MONDINO Abstract. For metric measure spaces satisfying the reduced curvature-dimension condition CD, N) we prove a series of sharp functional inequalities under the additional assumption of essentially nonbranching. Examples of spaces entering this framework are weighted) iemannian manifolds satisfying lower icci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and more generally CD, N)-spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower icci curvature bounds. In particular we prove the Brunn-Minkowski inequality, the p-spectral gap or equivalently the p- Poincaré inequality) for any p [1, ), the log-sobolev inequality, the Talagrand inequality and finally the Sobolev inequality. All the results are proved in a sharp form involving an upper bound on the diameter of the space; all our inequalities for essentially non-branching CD, N) spaces take the same form as the corresponding sharp ones known for a weighted iemannian manifold satisfying the curvature-dimension condition CD, N) in the sense of Bakry-Émery. In this sense inequalities are sharp. We also discuss the rigidity and almost rigidity statements associated to the p-spectral gap. Finally let us mention that for essentially non-branching metric measure spaces, the local curvaturedimension condition CD loc, N) is equivalent to the reduced curvature-dimension condition CD, N). Therefore we also have shown that the sharp Brunn-Minkowski inequality in the global form can be deduced from the local curvature-dimension condition, providing a step towards the long-standing problem of) globalization for the curvature-dimension condition CD, N). To conclude, some of the results can be seen as answers to open problems proposed in the Optimal Transport book of Villani [75]. 1. Introduction The theory of metric measure spaces satisfying a synthetic version of lower curvature and upper dimension bounds is nowadays a rich and well-established theory; nevertheless some important functional and geometric inequalities are in some cases still not proven and in others not proven in a sharp form. The scope of this note is to generalize several functional inequalities known for iemannian manifolds satisfying a lower bound on the icci curvature to the more general case of metric measure spaces satisfying the socalled curvature-dimension condition CD, N) as defined by Lott-Villani [51] and Sturm [72, 73]. More precisely our results will hold under the reduced curvature dimension condition CD, N) introduced by Bacher-Sturm [7] which is, a priori, a weaker assumption than the classic CD, N)) coupled with an essentially non-branching assumption on geodesics. We refer to Section 2.1 for the precise definitions; here let us recall that remarkable examples of essentially non-branching CD, N) spaces are weighted) iemannian manifolds satisfying lower icci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and more generally CD, N)-spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower icci curvature bounds. emark 1.1. To avoid technicalities in the introduction, all the results will be stated for N > 1; nevertheless everything holds and will be proved in the paper) also for N = 1, but in this case CD, N) has to be replaced by CD loc, N). The two conditions are equivalent for N > 1 and for N = 1, 0, but in case N = 1, < 0 the CD loc, N) condition is strictly stronger see Section 2.1 for more details). Before committing a paragraph to each of the functional inequalities we will consider in this note, we underline that most of the proofs contained in this note are based on L 1 optimal transportation theory and in particular on one-dimensional localization. This technique, having its roots in a work ey words and phrases. optimal transport; icci curvature lower bounds; metric measure spaces; Brunn-Minkowski inequality; log-sobolev inequality; spectral gap; Sobolev inequality; Talagrand inequality. 1

2 2 FABIO CAVALLETTI AND ANDEA MONDINO of Payne-Weinberger [64] and developed by Gromov-Milman [38], Lovász-Simonovits [53] and annan- Lovász-Simonovits [42], consists in reducing an n-dimensional problem to a one dimensional one via tools of convex geometry. ecently lartag [45] found an L 1 -optimal transportation approach leading to a generalization of these ideas to iemannian manifolds; the authors [17], via a careful analysis avoiding any smoothness assumption, generalized this approach to metric measure spaces. It is also convenient to introduce here the family of one-dimensional measures that will be used several times for comparison: F := { µ P): suppµ) [0, D], µ = h µ L 1, h µ C 2 0, D)),,, µ) CD, N) }, where,, µ) CD, N) stands for: the metric measure space,, µ) verifies CD, N) or equivalently ) h 1 µ + N 1 h 1 µ Brunn-Minkowski inequality. The celebrated Brunn-Minkowski inequality estimates from below the measure of the t-intermediate points between two given subsets A 0 and A 1 of, for t [0, 1]. For metric measure spaces satisfying the reduced curvature-dimension condition CD, N) see Section 2.1 for a brief account of different versions of the curvature-dimension condition) almost by definition for any A 0, A 1 1.1) ma t ) 1/N σ 1 t) θ) ma 0) 1/N + σ t) θ) ma 1) 1/N, where A t is the set of t-intermediate points between A 0 and A 1, that is ) A t = e t {γ Geo): γ 0 A 0, γ 1 A 1 }, see Section 2 for the definition of e) θ is the minimal/maximal length of geodesics from A 0 to A 1 : { inf x0,x θ := 1) A 0 A 1 dx 0, x 1 ), if 0, sup x0,x 1) A 0 A 1 dx 0, x 1 ), if < 0, and σ t) θ) is defined in 2.3). Nevertheless 1.1) is not sharp. Indeed if, d, m) is a weighted iemannian manifold satisfying CD, N), then 1.1) holds but with better interpolation coefficients, that is with τ t) 1 t) θ), τ θ) replacing σt) θ) and σ1 t) θ), respectively. Indeed for a weighted iemannian manifold the two a priori) different definitions of CD, N) and CD, N) coincide and then again almost by definition [73] one can obtain the improved and sharp) Brunn-Minkowski inequality let us mention that a direct proof of the Brunn-Minkowski inequality in the smooth setting was done earlier by Cordero-Erausquin, McCann and Schmuckenschläger [27]). A first main result of this paper is to establish the sharp inequality for essentially non-branching CD, N) metric measure spaces. Theorem 1.2 Theorem 3.1). Let, d, m) with m) < verify CD, N) for some, N and N 1, ). Assume moreover, d, m) to be essentially non-branching. Then it satisfies the following sharp Brunn-Minkowski inequality: for any A 0, A 1 ma t ) 1/N τ 1 t) θ) ma 0) 1/N + τ t) θ) ma 1) 1/N, where A t is the set of t-intermediate points between A 0 and A 1 and θ the minimal/maximal length of geodesics from A 0 to A 1. emark 1.3. The remarkable feature of Theorem 1.2 is that the sharp Brunn-Minkowski inequality in the global form can be deduced from the local curvature-dimension condition, providing a step towards the long-standing problem of) globalization for the curvature-dimension condition CD, N). For an account and for partial results about this problem we refer to [6, 7, 16, 18, 68, 75].

3 SHAP GEOMETIC AND FUNCTIONAL INEQUALITIES FO CD, N) SPACES p-spectral gap. In the smooth setting, a spectral gap inequality establishes a bound from below on the first eigenvalue of the Laplacian. More generally, for any p 1, ) one can define the positive real number,d,m) as follows { },d,m) := inf f p m f p m : f Lip) Lp, m), f 0, f f p 2 m = 0, where f is the slope also called local Lipshitz constant) of the Lipschitz function f. The name is motivated by the fact that in case, d, m) is the m.m.s corresponding to a smooth compact iemannian manifold then,d,m) coincides with the first positive eigenvalue of the problem p f = λ f p 2 f, on, d, m), where p f := div f p 2 f) is the so called p-laplacian. We now state the main theorem of this paper on p-spectral gap inequality. Theorem 1.4 Theorem 4.4). Let, d, m) be a metric measure space satisfying CD, N) for some, N and N 1, ) and assume moreover it is essentially non-branching. Let D 0, ) be the diameter of. Then for any p 1, ) it holds,d,m), where is defined by := inf inf µ F { u p µ u p µ : u Lip) Lp µ), } u u p 2 µ = 0, u 0. In other terms for any Lipschitz function f L p, m) with f f p 2 mdx) = 0 it holds fx) p mdx) f p x) mdx). For more about the quantity the reader is referred to Section 4.1 where the model spaces are discussed in detail. From the last formulation of the statement, it is clear that the sharp p-spectral gap above is equivalent to a sharp p-poincaré inequality. Let us now give a brief and incomplete) account on the huge literature about the spectral gap. When the ambient metric measure space is a smooth iemannian manifold equipped with the volume measure, the study of the first eigenvalue of the Laplace-Beltrami operator has a long history going back to Lichnerowicz [50], Cheeger [21], Li-Yau [49], etc. For an overview the reader can consult for instance the book by Chavel [20], the survey by Ledoux [47], or Chapter 3 in Shoen-Yau s book [71], and references therein. We mention that the estimate of Theorem 1.4 in the case p = 2 started with Payne-Weinberger [64] for convex domains in n where diameter-improved spectral gap inequality for the Laplace operator was originally proved. Later it was generalized to iemannian manifolds with non-negative icci curvature by Yang-Zhong [77], and by Bakry-Qian [9] for manifolds with densities. The generalization to arbitrary p 1, ) has been proved by Valtorta [74] for = 0 and Naber-Valtorta [61] for any. All of these results hold for iemannian manifolds. egarding metric measure spaces, the sharp Lichnerowitz spectral gap for p = 2 was proved by Lott- Villani [52] under the CD, N) condition. Jiang-Zhang [41] recently showed, still for p = 2, that the improved version under an upper diameter bound holds for CD, N) metric measure spaces. For icci limit spaces, in the case > 0 and D = π N 1)/, the p-spectral gap above has been recently obtained by Honda [40] via proving the stability of under mgh convergence of compact iemannian manifolds; this approach was inspired by the celebrated work of Cheeger-Colding [25] where, in particular, it was shown the stability of λ 1,2 under mgh convergence. We also obtain the almost rigidity for the p- spectral gap: if an almost equality in the p-spectral gap holds, then the space must have almost maximal diameter. Theorem 1.5 Theorem 4.5). Let N > 1, and p 1, ) be fixed. Then for every ε > 0 there exists δ = δε, N, p) such that the following holds.

4 4 FABIO CAVALLETTI AND ANDEA MONDINO Let, d, m) be an essentially non-branching metric measure space satisfying CD N 1 δ, N + δ). If,d,m) λ1,p,n,π + δ, then diam ) π ε. As a consequence, by a compactness argument and using the Maximal Diameter Theorem proved recently for CD, N) by etterer [43], we have the following p-obata and almost p-obata Theorems. Corollary 1.6 p-obata Theorem). Let, d, m) be an CD N 1, N) space for some N 2, and let 1 < p <. If,d,m) = λ1,p,n,π = λ1,p S N )), then, d, m) is a spherical suspension, i.e. there exists an CD N 2, N 1) space Y, d Y, m Y ) such that, d, m) is isomorphic to [0, π] sin Y. Corollary 1.7 Almost p-obata Theorem). Let N 2, and p 1, ) be fixed. Then for every ε > 0 there exists δ = δε, N, p) > 0 such that the following holds. Let, d, m) be an CD N 1 δ, N + δ) space. If,d,m) λ1,p,n,π + δ, then there exists an CD N 2, N 1) space Y, d Y, m Y ) such that d mgh, d, m), [0, π] sin Y ) ε. Let us mention that the classical Obata s Theorem for CD, N)-spaces, i.e. the version of Corollary 1.6 for p = 2, was recently obtained by etterer [44] see also [41]) with different methods. Finally we recall that the case p = 1 can be attacked using the identity h,d,m) = λ 1,1,d,m), where h,d,m) is the so-called Cheeger isoperimetric constant, see Section 5.1. Therefore Theorem 1.4, Theorem 1.5, Corollary 1.6 and Corollary 1.7 for the case p = 1 follow from the analogous results proved for the isoperimetric profile in [17]. Nevertheless for reader s convenience, the case p = 1 will be discussed in detail in Section Log-Sobolev and Talagrand inequality. Given a m.m.s., d, m), we say that it supports the Log-Sobolev inequality with constant α > 0 if for any Lipschitz function f : [0, ) with fx) mdx) = 1 it holds f 2 1.2) 2α f log f m m. {f>0} f The largest constant α, such that 1.2) holds for any Lipschitz function f : [0, ) with fx) mdx) = 1, will be called Log-Sobolev constant of, d, m) and denoted with α,d,m) LS. Log-Sobolev inequality is already known [75, Theorem 30.22] for essentially non-branching metric measure spaces satisfying CD, ) with > 0 with sharp constant α =, but it is an open problem see for instance [75, Open Problem 21.6]) to get the sharp dimensional constant α = N for metric measure spaces with N-icci curvature bounded below by. This is the goal of the next result. As already done above, let us introduce the model constant for the one-dimensional case. Given, N 1, D 0, + ) we denote with α LS > 0 the maximal constant α such that f 2 1.3) 2 α f log f µ f µ, µ F, {f>0} for every Lipschitz f : [0, ) with f µ = 1. emark 1.8. If > 0 and D = π constant is N, it is known that the corresponding optimal Log-Sobolev for more details see the discussion in Section 6.1). It is an interesting open problem, that we don t address here, to give an explicit expression of the quantity α LS D 0, ). for general, N 1, Theorem 1.9 Sharp Log-Sobolev inequality, Theorem 6.2). Let, d, m) be a metric measure space with diameter D 0, ) and satisfying CD, N) for some, N 1, ). Assume moreover it is essentially non-branching.

5 SHAP GEOMETIC AND FUNCTIONAL INEQUALITIES FO CD, N) SPACES 5 Then for any Lipschitz function f : [0, ) with f m = 1 it holds 2 α LS f 2 f log f m m. f In other terms it holds α LS,d,m) αls. {f>0} As a consequence, if > 0 and no diameter upper bound is assumed or D = π i.e. for any Lipschitz function f : [0, ) with f m = 1 it holds 2N f 2 f log f m m. N 1 f {f>0}, then αls = N In order to state the Talagrand inequality let us recall that the relative entropy functional Ent m : P) [0, + ] with respect to a given m P) is defined to be Ent m µ) = ϱ log ϱ m, if µ = ϱm and + otherwise. Otto-Villani [62] proved that for smooth iemannian manifolds the Log-Sobolev inequality with constant α > 0 implies the Talagrand inequality with constant 2 α preserving sharpness. The result was then generalized to arbitrary metric measure spaces by Gigli-Ledoux [34]. Combining this result with Theorem 1.9 we get the following corollary which improves the Talagrand constant 2/, which is sharp for CD, ) spaces, by a factor N 1/N in case the dimension is bounded above by N. This constant is sharp for CD, N) or CD loc, N)) spaces, indeed it is sharp already in the smooth setting [75, emark 22.43]. Since both our proof of the sharp Log-Sobolev inequality and the proof of Theorem 6.4 are essentially optimal transport based, the following can be seen as an answer to [75, Open Problem 22.44]. Theorem 1.10 Sharp Talagrand inequality). Let, d, m) be a metric measure space with diameter D 0, ), satisfying CD, N) for some, N 1, ), and assume moreover it is essentially non-branching and m) = 1. 2 Then it supports the Talagrand inequality with constant, where α α LS LS was defined in 1.3), i.e. it holds W 2 2 µ, m) 2 α LS Ent m µ) for all µ P). In particular, if > 0 and no upper bound on the diameter is assumed or D = π W2 2 2N 1) µ, m) N Ent mµ) for all µ P), the constant in the last inequality being sharp., then 1.4. Sobolev inequality. Sobolev inequalities have been studied in many different contexts and many papers and books are devoted to this family of inequalities. Here we only mention two references mainly dealing with them in the iemannian manifold case and the smooth CD condition case, respectively [39] and [46]. We say that, d, m) supports a p, q)-sobolev inequality with constant α p,q if for any f : Lipschitz function it holds { α p,q ) q } 1.4) f p p m f q m f q m, p q and the largest constant α p,q such that 1.4) holds for any Lipschitz function f will be called the p, q)- Sobolev constant of, d, m)and will be denoted by α p,q,d,m). A Sobolev inequality is known to hold for essentially non-branching m.m.s. satisfying CD, N), provided < 0, see [75, Theorem 30.23] and other Sobolev-type inequalities have been obtained in [52] for CD, N) spaces. Let us also mention [66] where the sharp 2, 2)-Sobolev inequality has been established for CD, N)-spaces, > 0, N 2, ). The goal here is to give a Sobolev inequality with sharp constant for essentially non-branching CD, N) spaces,, N > 1, taking also into account an upper diameter bound.

6 6 FABIO CAVALLETTI AND ANDEA MONDINO Theorem 1.11 Sharp Sobolev inequality, Theorem 7.1). Let, d, m) be a metric measure space with diameter D 0, ) and satisfying CD, N) for some, N 1, ). Assume moreover it is essentially non-branching. Then for any Lipschitz function it holds α p,q p q { fx) p mdx) ) q p } fx) q mdx) fx) q mdx), where α p,q is defined as the supremum among α > 0 such that { ) q } α f p p µ f q µ f q µ, f Lip, µ F. p q In particular, if > 0, N > 2 and no upper bound on the diameter is assumed or D = π any Lipschitz function f it holds { ) 2 } N f p p m f 2 m f 2 m, p 2)N 1) for any 2 < p 2N/N 2); in other terms it holds α p,2,d,m) N. This last result can be seen as a solution to [75, Open Problem 21.11]. Acknowledgements, then for The authors wish to thank the Hausdorff center of Mathematics of Bonn, where part of the work has been developed, for the excellent working conditions and the stimulating atmosphere during the trimester program Optimal Transport. A.M. is partly supported by the Swiss National Science Foundation. 2. Prerequisites In what follows we say that a triple, d, m) is a metric measure space, m.m.s. for short, if, d) is a complete and separable metric space and m is positive adon measure over. For this note we will only be concerned with m.m.s. with m probability measure, that is m) = 1, or at most with m) < which will be reduced to the probability case by a constant rescaling. The space of all Borel probability measure over will be denoted with P). A metric space is a geodesic space if and only if for each x, y there exists γ Geo) so that γ 0 = x, γ 1 = y, with Geo) := {γ C[0, 1], ) : dγ s, γ t ) = s t)dγ 0, γ 1 ), s, t [0, 1]}. ecall that for complete geodesic spaces local compactness is equivalent to properness a metric space is proper if every closed ball is compact). We directly assume the ambient space, d) to be proper. Hence from now on we assume the following: the ambient metric space, d) is geodesic, complete, separable and proper and m) = 1. We denote with P 2 ) the space of probability measures with finite second moment endowed with the L 2 -Wasserstein distance W 2 defined as follows: for µ 0, µ 1 P 2 ) we set 2.1) W2 2 µ 0, µ 1 ) = inf d 2 x, y) πdxdy), π where the infimum is taken over all π P ) with µ 0 and µ 1 as the first and the second marginal. Assuming the space, d) to be geodesic, also the space P 2 ), W 2 ) is geodesic. Any geodesic µ t ) t [0,1] in P 2 ), W 2 ) can be lifted to a measure ν PGeo)), so that e t ) ν = µ t for all t [0, 1]. Here for any t [0, 1], e t denotes the evaluation map: e t : Geo), e t γ) := γ t. Given µ 0, µ 1 P 2 ), we denote by OptGeoµ 0, µ 1 ) the space of all ν PGeo)) for which e 0, e 1 ) ν realizes the minimum in 2.1). If, d) is geodesic, then the set OptGeoµ 0, µ 1 ) is non-empty for any µ 0, µ 1 P 2 ). It is worth also introducing the subspace of P 2 ) formed by all those measures absolutely continuous with respect to m: it is denoted by P 2, d, m).

7 SHAP GEOMETIC AND FUNCTIONAL INEQUALITIES FO CD, N) SPACES Geometry of metric measure spaces. Here we briefly recall the synthetic notions of lower icci curvature bounds, for more detail we refer to [7, 51, 72, 73, 75]. In order to formulate curvature properties for, d, m) we introduce the following distortion coefficients: given two numbers, N with N 1, we set for t, θ) [0, 1] +,, if θ 2 Nπ 2, sintθ /N) 2.2) σ t) θ) := sinθ if 0 < θ 2 < Nπ 2, /N) t if θ 2 < 0 and N = 0, or if θ 2 = 0, sinhtθ /N) sinhθ if θ 2 0 and N > 0. /N) We also set, for N 1, and t, θ) [0, 1] + 2.3) τ t) θ) := t1/n σ t), θ))/n. As we will consider only the case of essentially non-branching spaces, we recall the following definition. Definition 2.1. A metric measure space, d, m) is essentially non-branching if and only if for any µ 0, µ 1 P 2 ) which are absolutely continuous with respect to m any element of OptGeoµ 0, µ 1 ) is concentrated on a set of non-branching geodesics. A set F Geo) is a set of non-branching geodesics if and only if for any γ 1, γ 2 F, it holds: t 0, 1) : γ 1 t = γ 2 t, t 0, t) = γ 1 s = γ 2 s, s [0, 1]. Definition 2.2 CD condition). An essentially non-branching m.m.s., d, m) verifies CD, N) if and only if for each pair µ 0, µ 1 P 2, d, m) there exists ν OptGeoµ 0, µ 1 ) such that 2.4) ϱ 1/N t γ t ) τ 1 t) dγ 0, γ 1 ))ϱ 1/N 0 γ 0 ) + τ t) dγ 0, γ 1 ))ϱ 1/N 1 γ 1 ), ν-a.e. γ Geo), for all t [0, 1], where e t ν = ϱ t m. For the general definition of CD, N) see [51, 72, 73]. It is worth recalling that if M, g) is a iemannian manifold of dimension n and h C 2 M) with h > 0, then the m.m.s. M, g, h vol) verifies CD, N) with N n if and only if see Theorem 1.7 of [73]) ic g,h,n g, ic g,h,n := ic g N n) 2 gh 1 N n h 1 N n In particular if N = n the generalized icci tensor ic g,h,n = ic g makes sense only if h is constant. In particular, if I is any interval, h C 2 I) and L 1 is the one-dimensional Lebesgue measure, the m.m.s. I,, hl 1 ) verifies CD, N) if and only if 2.5) h 1 ) + N 1 h 1 0, and verifies CD, 1) if and only if h is constant. We also mention the more recent iemannian curvature dimension condition CD introduced in the infinite dimensional case in [4, 2, 1] and in the finite dimensional case in [28, 5]. We refer to these papers and references therein for a general account on the synthetic formulation of icci curvature lower bounds for metric measure spaces. Here we only mention that CD, N) condition is an enforcement of the so called reduced curvature dimension condition, denoted by CD, N), that has been introduced in [7]: in particular the additional condition is that the Sobolev space W 1,2, m) is an Hilbert space, see [3, 4]. The reduced CD, N) condition asks for the same inequality 2.4) of CD, N) but the coefficients τ t) dγ 0, γ 1 )) and τ 1 t) dγ 0, γ 1 )) are replaced by σ t) dγ 0, γ 1 )) and σ 1 t) dγ 0, γ 1 )), respectively. Hence while the distortion coefficients of the CD, N) condition are formally obtained imposing one direction with linear distortion and N 1 directions affected by curvature, the CD, N) condition imposes the same volume distortion in all the N directions. It was proved in [69] that the CD, N) condition implies the essentially non-branching property, so this is a fairly natural assumption in the framework of m.m.s. satisfying lower icci bounds..

8 8 FABIO CAVALLETTI AND ANDEA MONDINO For both CD-CD definitions there is a local version that is of some relevance for our analysis. Here we state only the local formulation CD, N), the one for CD, N) being similar. Definition 2.3 CD loc condition). An essentially non-branching m.m.s., d, m) satisfies CD loc, N) if for any point x there exists a neighborhood x) of x such that for each pair µ 0, µ 1 P 2, d, m) supported in x) there exists ν OptGeoµ 0, µ 1 ) such that 2.4) holds true for all t [0, 1]. The support of e t ν is not necessarily contained in the neighborhood x). One of the main properties of the reduced curvature dimension condition is the globalization one: under the essentially non-branching property, CD loc, N) and CD, N) are equivalent see [7, Corollary 5.4]). Let us mention that the local-to-global property is satisfied also by the CD, N) condition, see [6]. We also recall few relations between CD and CD. It is known by [32, Theorem 2.7] that, if, d, m) is a non-branching metric measure space satisfying CD, N) and µ 0, µ 1 P) with µ 0 absolutely continuous with respect to m, then there exists a unique optimal map T : such that id, T ) µ 0 realizes the minimum in 2.1) and the set OptGeoµ 0, µ 1 ) contains only one element. The same proof holds if one replaces the non-branching assumption with the more general one of essentially non-branching, see for instance [69]. emark 2.4 CD, N) Vs CD loc, N)). esults of [7] imply the following chain of implications: if, d, m) is a proper, essentially non-branching, metric measure space, then CD loc, N) CD loc, N) CD, N), provided, N with N > 1 or N = 1 and 0. Let us remark that on the other hand CD, 1) does not imply CD loc, 1) for < 0: indeed it is possible to check that, d, m) = [0, 1],, c sinh )L 1 ) satisfies CD 1, 1) but not CD loc 1, 1) which would require the density to be constant. Hence CD, N) and CD loc, N) are equivalent if 1 < N < or N = 1 and 0, but for N = 1 and < 0 the CD loc, N) condition is strictly stronger than CD, N). Note also that many results presented in [7] are for metric measure spaces verifying CD, N) and its local version), that is they verify the CD, N) condition for all <. Thanks to uniqueness of geodesics in P 2 ), W 2 ) guaranteed by the essentially non-branching assumption, CD, N) is equivalent to CD, N). As a final comment we also mention that, for > 0, CD, N) implies CD, N) where = N 1)/N. For a deeper analysis on the interplay between CD and CD we refer to [16, 18] Measured Gromov-Hausdorff convergence and stability of CD, N). Let us first recall the notion of measured Gromov-Hausdorff convergence, mgh for short. Since in this work we will apply it to compact m.m. spaces endowed with probability measures having full support, we will restrict to this framework for simplicity for a more general treatment see for instance [35]). Definition 2.5. A sequence j, d j, m j ) of compact m.m. spaces with m j j ) = 1 and suppm j ) = j is said to converge in the measured Gromov-Hausdorff topology mgh for short) to a compact m.m. space, d, m ) with m ) = 1 and suppm ) = if and only if there exists a separable metric space Z, d Z ) and isometric embeddings {ι j :, d j ) Z, d Z )} i N such that for every ε > 0 there exists j 0 such that for every j > j 0 ι ) B Z ε [ι j j )] and ι j j ) B Z ε [ι )], where Bε Z [A] := {z Z : d Z z, A) < ε} for every subset A Z, and ϕ ι j ) m j )) ϕ ι ) m )) Z Z ϕ C b Z), where C b Z) denotes the set of real valued bounded continuous functions in Z. The following theorem summarizes the compactness/stability properties we will use in the proof of the almost rigidity result notice these hold more generally for every by replacing mgh with pointed-mgh convergence). Theorem 2.6 Metrizability and Compactness). Let > 0, N > 1 be fixed. Then the mgh convergence restricted to isomorphism classes of) CD, N) spaces is metrizable by a distance function d mgh. Furthermore every sequence j, d j, m j ) of CD, N) spaces admits a subsequence which mgh-converges to a limit CD, N) space.

9 SHAP GEOMETIC AND FUNCTIONAL INEQUALITIES FO CD, N) SPACES 9 The compactness follows by the standard argument of Gromov, indeed for fixed > 0, N > 1, the spaces have uniformly bounded diameter, moreover the measures of CD, N) spaces are uniformly doubling, hence the spaces are uniformly totally bounded and thus compact in the GH-topology; the weak compactness of the measures follows using the doubling condition again and the fact that they are normalized. For the stability of the CD, N) condition under mgh convergence see for instance [7, 28, 35]. The metrizability of mgh convergence restricted to a class of uniformly doubling normalized m.m. spaces having uniform diameter bounds is also well known, see for instance [35] Warped product. Given two geodesic m.m.s. B, d B, m b ) and F, d F, m F ) and a Lipschitz function f : B + one can define a length function on the product B F : for any absolutely continuous curve γ : [0, 1] B F with γ = α, β), define 1 Lγ) := α 2 t) + f α) 2 t) β 1/2 t)) 2 dt and define accordingly the pseudo-distance Then the warped product of B with F is defined as 0 p, x), q, y) := inf {Lγ): γ 0 = p, x), γ 1 = q, y)}. B f F := B F /,, ), where p, x) q, y) if and only if p, x), q, y) = 0. One can also associate a measure and obtain the following object B N f F := B f F, m C ), m C := f N m B m F. Then B N f F will be a metric measure space called measured warped product. For a general picture on the curvature properties of warped products, we refer to [43] Localization method. The next theorem represents the key technical tool of the present paper. The roots of such a result, known in literature as localization technique, can be traced back to a work of Payne-Weinberger [64] further developed in the Euclidean space by Gromov-Milman [38], Lovász- Simonovits [53] and annan-lovász-simonovits [42]. The basic idea consists in reducing an n-dimensional problem to a one dimensional one via tools of convex geometry. ecently lartag [45] found an L 1 -optimal transportation approach leading to a generalization of these ideas to iemannian manifolds; the authors [17], via a careful analysis avoiding any smoothness assumption, generalized this approach to metric measure spaces. Theorem 2.7. Let, d, m) be an essentially non-branching metric measure space with m) = 1 satisfying CD loc, N) for some, N and N [1, ). Let f : be m-integrable such that f m = 0 and assume the existence of x 0 such that fx) dx, x 0) mdx) <. Then the space can be written as the disjoint union of two sets Z and T with T admitting a partition { q } q Q, where each q is the image of a geodesic; moreover there exists a family of probability measures {m q } q Q P) with the following properties: For any m-measurable set B T it holds mb) = m q B) qdq), where q is a probability measure over Q. Q For q-almost every q Q, the set q is a geodesic with strictly positive length and m q is supported on it. Moreover q m q is a CD, N) disintegration, that is m q = gq, ) h q L 1), with 2.6) h q 1 s)t 0 + st 1 ) 1 1 s) σ, t 1 t 0 )h q t 0 ) 1 s) + σ, t 1 t 0 )h q t 1 ) 1, for all s [0, 1] and for all t 0, t 1 Dom gq, )) with t 0 < t 1, where gq, ) is the isometry with range q. If N = 1, for q-a.e. q Q the density h q is constant. For q-almost every q Q, it holds q f m q = 0 and f = 0 m-a.e. in Z.

10 10 FABIO CAVALLETTI AND ANDEA MONDINO emark 2.8. Inequality 2.6) is the weak formulation of the following differential inequality on h q,t0,t 1 : ) 2.7) h 1 q,t 0,t 1 + t 1 t 0 ) 2 N 1 h 1 q,t 0,t 1 0, for all t 0 < t 1 Dom gq, )), where h q,t0,t 1 s) := h q 1 s)t 0 + st 1 ). It is easy to observe that the differential inequality 2.7) on h q,t0,t 1 is equivalent to the following differential inequality on h q : ) h 1 q + N 1 h 1 q 0, that is precisely 2.5). Then Theorem 2.7 can be alternatively stated as follows. If, d, m) is an essentially non-branching m.m.s. verifying CD loc, N) and ϕ : is a 1- Lipschitz function, then the corresponding decomposition of the space in maximal rays { q } q Q produces a disintegration {m q } q Q of m so that for q-a.e. q Q, the m.m.s. Dom gq, )),, h q L 1 ) verifies CD, N). Accordingly, from now on we will say that the disintegration q m q is a CD, N) disintegration. Few comments on Theorem 2.7 are in order. From 2.6) it follows that 2.8) {t Dom gq, )): h q t) > 0} is convex and t h q t) is locally Lipschitz continuous. The measure q is the quotient measure associated to the partition { q } q Q of T and Q its quotient set, see [17] for details. 3. Sharp Brunn-Minkowski inequality In this section we prove sharp Brunn-Minkowski inequality for m.m.s. satisfying CD loc, N). It follows from emark 2.4 that the same result holds under CD, N) for any, N, provided N 1, ) or N = 1 and 0. See also emark 1.1. The same will hold for all the inequalities proved in the paper. Theorem 3.1. Let, d, m) with m) < verify CD loc, N) for some N, and N [1, ). Assume moreover, d, m) to be essentially non-branching. Then it satisfies the following sharp Brunn- Minkowski inequality: for any A 0, A 1 3.1) ma t ) 1/N τ 1 t) θ)ma 0) 1/N + τ t) θ)ma 1) 1/N, where A t is the set of t-intermediate points between A 0 and A 1, that is ) A t = e t {γ Geo): γ 0 A 0, γ 1 A 1 }, and θ the minimal/maximal length of geodesics from A 0 to A 1 : { inf x0,x θ := 1) A 0 A 1 dx 0, x 1 ), if 0, sup x0,x 1) A 0 A 1 dx 0, x 1 ), if < 0. Before starting the proof of Theorem 3.1 we recall the classical result of Borell [11] and Brascamp-Lieb [12] characterizing one-dimensional measures satisfying Brunn-Minkowski inequality. Lemma 3.2. Let η be a Borel measure defined on admitting the following representation: η = h L 1. The following are equivalent: i) The density h is, N)-concave on its convex support, that is ) h 1 + N 1 h 1 0, in the weak sense, see 2.6).

11 SHAP GEOMETIC AND FUNCTIONAL INEQUALITIES FO CD, N) SPACES 11 ii) For any A 0, A 1 subsets of ηa t ) τ 1 t) θ) ηa 0) 1/N + τ t) θ) ηa 1) 1/N, where A t := {1 t)x + ty : x A 0, y A 1 } and θ is the minimal/maximal length of geodesics from A 0 to A 1 : { ess inf x0,x θ := 1) A 0 A 1 dx 0, x 1 ), if 0, ess sup x0,x 1) A 0 A 1 dx 0, x 1 ), if < 0. For reader s convenience we include here a proof that i) implies ii), which is the implication we will use later. Proof. Consider the N-entropy: for any µ = ρ η S N µ η) := ρ 1/N x) µdx). Observe that ii) is implied by displacement convexity of S N with respect to the L 2 -Wasserstein distance over, ). Just consider µ 0 := ηa 0 ) 1 η A0 and µ 1 := ηa 1 ) 1 η A1 and use Jensen s inequality. Consider therefore a geodesic curve [0, 1] t ρ t η W 2, ), T t ρ 0 η = ρ t η, where T t = Id1 t) + tt and T is the µ 0 -essentially) unique monotone rearrangement such that T µ 0 = µ 1. Thanks to approximate differentiability of T, one can use change of variable formula and obtain the following chain of equalities: ρ t x) N ηdx) = suppµ t) ρ t T t x)) ht t x)) 1 t) + tt x) = ρ 0 x)hx) = = suppµ t) suppµ 0) suppµ 0) ρ t x) N hx) dx ρ t T t x)) N htt x)) 1 t) + tt dx ρ 0 x) N htt x)) ) 1 N 1 t) + tt x) 1 N ηdx). hx) Hence the claim has became to prove that t J t x) 1 N is concave, where J t is the Jacobian of T t with respect to η and J t x) = Jt G x) Jt W x), Jt G x) = 1 t) + tt x), Jt W x) = ht tx)), hx) where J G is the geometric Jacobian and J W the weighted Jacobian. Since t Jt G x) is linear, using Hölder s inequality the claim follows straightforwardly from the, N)-convexity of h. We can now move to the proof of Theorem Proof of Theorem 3.1. First of all notice that up to replacing m with the normalized measure m) m we can assume that m) = 1. Let A 0, A 1 be two given Borel sets of positive m-measure. Step 1. Consider the function f := χ A0 /ma 0 ) χ A1 /ma 1 ) and observe that f m = 0. From Theorem 2.7, the space can be written as the disjoint union of two sets Z and T with T admitting a partition { q } q Q and a corresponding disintegration of m T, {m q } q Q such that: m T = m q qdq), Q where q is the quotient measure, for q-almost every q Q, the set q is a geodesic, m q is supported on it and q m q is a CD, N) disintegration. Finally, for q-almost every q Q, it holds q f m q = 0 and f = 0 m-a.e. in Z. We can also consider the trivial disintegration of m restricted to Z where each equivalence class is a single point: m Z = δ z mdz), Z

12 12 FABIO CAVALLETTI AND ANDEA MONDINO where δ z stands for the Dirac delta in z. Then define q := q + m Z and m q = m q if q Q and m q = δ q if q Z. Since Q Z =, the previous definitions are well posed and we have the following decomposition of m on the whole space m = m q qdq). Q Z Step 2. Use the following notation A 0,q := A 0 q, A 1,q := A 1 q and the set of t-intermediate points between A 0,q and A 1,q in q is denoted with A t,q q. Then from Lemma 3.2, for q-a.e. q Q m q A t,q ) Since fm q = 0 implies mqa0,q) ma 0) τ 1 t) 3.2) m q A t,q ) m qa 0,q ) ma 0 ) θ)m qa 0,q ) 1/N + τ t) θ)m qa 1,q ) 1/N ) N. = mqa1,q) ma 1), it follows that τ 1 t) θ)ma 0) 1/N + τ t) θ)ma 1) 1/N) N. We now show that 3.2) holds also for q-a.e. or equivalently m-a.e.) q Z. Note that in this case m q has to be replaced by δ q. Since by construction 0 = f = χ A0 /ma 0 ) χ A1 /ma 1 ) on Z, then necessarily m Z \ A 0 A 1 ) \ A 0 A 1 ) )) = 0. It follows that if Z does not have m-measure zero, we have two possibilities: m Z \ A 0 A 1 ))) > 0, or ma 0 ) = ma 1 ) and m Z A 0 A 1 )) > 0. Therefore, if mz) > 0, for q-a.e. or equivalently m-a.e.) q Z we have two possibilities: q \ A 0 A 1 ), or q A 0 A 1. Interpreting the intermediate points as the point itself, in the first case 3.2) with m q replaced by δ q ) holds trivially i.e. we get 0 0). In the second case it reduces to show that τ 1 t) ) N t) θ) + τ θ) 1. For 0, since we are in the case ma 0 A 1 ) > 0, it follows that θ = 0 and therefore τ t) θ) = t, proving the previous inequality. For < 0, recalling that σ t) θ) is non-decreasing see [7], emark 2.2 ), by Hölder s inequality τ 1 t) ) N t) θ) + τ θ) 1 t + t) as desired. We have therefore proved that 3.3) m q A t,q ) m qa 0,q ) ma 0 ) τ 1 t) σ 1 t), ) θ) + σt), θ) 1, θ)ma 0) 1/N + τ t) θ)ma 1) 1/N) N, for q-a.e. q Q Z. Taking the integral of 3.3) in q Q Z one obtains that ma t ) = m q A t q ) qdq) Q Z m q A t,q ) qdq) = Q Z τ 1 t) τ 1 t) θ)ma 0) 1/N + τ t) θ)ma 1) 1/N) N θ)ma 0) 1/N + τ t) θ)ma 1) 1/N) N, Q Z m q A 0,q ) ma 0 ) qdq) and the claim follows.

13 SHAP GEOMETIC AND FUNCTIONAL INEQUALITIES FO CD, N) SPACES p-spectral gap Given a metric space, d), we denote with Lip) respectively Lip c )) the vector space of real valued Lipschitz functions resp. with compact support). For a Lipschitz function f : the local Lipschitz constant f is defined by f x) = lim sup y x fx) fy) dx, y) if x is not isolated, 0 otherwise. For a m.m.s., d, m), for every p 1, ) we define the first eigenvalue, d, m) of the p- Laplacian by { } 4.1),d,m) := inf f p m f p m : f Lip) Lp, m), f 0, f f p 2 m = p-spectral gap for m.m.s. over, ): the model spaces. Consider the following family of probability measures F s := {µ P) : 4.2) suppµ) [0, D], µ = h µ L 1, h µ verifies 2.6) and is continuous if N 1, ), h µ const if N = 1}, where D 0, ) and the corresponding synthetic first non-negative eigenvalue of the p-laplacian { s := inf inf u p } µ µ F s u p µ : u Lip) Lp µ), u u p 2 µ = 0, u 0. The term synthetic refers to µ F s meaning that the icci curvature bound is satisfied in its synthetic formulation: if µ = h L 1, then h verifies 2.6). The first goal of this section is to prove that s defined by coincides with its smooth counterpart λ1,p { 4.3) := inf inf u p } µ µ F u p µ : u Lip) Lp µ), u u p 2 µ = 0, u 0, where now F denotes the set of µ P) such that suppµ) [0, D] and µ = h L 1 with h C 2 0, D)) satisfying ) 4.4) h 1 + N 1 h 1 0. It is easily verified that F F s. In order to prove that s = λ1,p the following approximation result, proved in [17, Lemma 6.2] will play a key role. In order to state it let us recall that a standard mollifier in is a non negative C ) function ψ with compact support in [0, 1] such that ψ = 1. Lemma 4.1. Let D 0, ) and let h : [0, D] [0, ) be a continuous function. Fix N 1, ) and for ε > 0 define [ [ 4.5) h ε t) := [h 1 ψε t)] := ht s) 1 ψε s) ds] = hs) 1 ψε t s) ds], where ψ ε x) = 1 ε ψx/ε) and ψ is a standard mollifier function. The following properties hold: 1) h ε is a non-negative C function with support in [ ε, D + ε]; 2) h ε h uniformly as ε 0, in particular h ε h in L 1. 3) If h satisfies the convexity condition 2.6) corresponding to the above fixed N > 1 and some then also h ε does. In particular h ε satisfies the differential inequality 4.4). Proposition 4.2. For every p 1, + ), N [1, ),, D 0, ) it holds s = λ1,p.

14 14 FABIO CAVALLETTI AND ANDEA MONDINO Proof. First of all observe that for N = 1 clearly we have F = F s since the density h µ has to be constant. We can then assume without loss of generality that N 1, ). Since F F s then clearly s λ1,p. Assume by contradiction the inequality is strict. Then there exists a measure µ = h L 1 F s and δ > 0 such that,,µ) λ1,p 2δ. Therefore, by the very definition of,,µ), there exists u Lip), such that u 0, u u p 2 h ds = 0 and 4.6) u s) p hs) ds 3 ) 2 δ us) p hs) ds. Now, Lemma 4.1 gives a sequence h k C ) such that [ 4.7) supph k ) 1 k, D + 1 ], µ k := h k L 1 F k + 2, k h k h uniformly on [0, D]. Called now u k := u c k Lip) L p, h k L 1 ) where c k are such that u k u k p 2 h k ds = 0, thanks to 4.7) it holds c k 0 and thus u k s) p h k s) ds us) p hs) ds and u ks) p h k s) ds u s) p hs) ds. Therefore 4.6), combined with the continuity of ε +ε see Theorem 4.3 below), implies that for k large enough one has u ks) p h k s) ds δ) contradicting the definition of + 2 k u k s) p h k s) ds δ ) u + 2 k s) p h k s) ds, k 2 given in 4.3). The next goal of the section is to understand the quantity. Since now the density of the reference probability measure is smooth, we enter into a more classical framework where a number of people contributed. The sharp p-spectral gap in case > 0 and without upper bounds on the diameter was obtained by Matei [54]. The case = 0 and the diameter is bounded above was obtained in the sharp form by Valtorta [74]. Finally the case < 0 and diameter bounded above was obtained in the sharp form by Naber-Valtorta [61]. Actually, as explained in their paper, the arguments in [61] hold in the general case, N [1, ), provided one identifies the correct model space. As usual, to describe the model space one has to examine separately the cases < 0, = 0 and > 0; in order to unify the presentation let us denote with tan t) the following function: /N 1) tanh /N 1)t) if < 0, 4.8) tan t) := 0 if = 0 /N 1) tan /N 1)t) if > 0. Now, for each, N [1, ), D 0, ), let [ D/2, D/2] of the eigenvalue problem 4.9) d dt λ 1,p ẇ p 1)) + N 1) tan t) ẇ p 1) 1,p + λ wp 1) = 0. λ 1,p denote the first positive eigenvalue on It is possible to show see [61]) that is the unique value of λ such that the solution of ) 1/p φ = λ p 1 + p 1 tan t) cos p 1) p φ) sin p φ) φ0) = 0 satisfies φd/2) = π p /2, where π p, cos p and sin p are defined as follows. For every p 1, ) the positive number π p is defined by π p := 1 1 ds 1 s p ) 1/p = 2π p sinπ/p).

15 SHAP GEOMETIC AND FUNCTIONAL INEQUALITIES FO CD, N) SPACES 15 The C 1 ) function sin p : [ 1, 1] is defined implicitly on [ π p /2, 3π p /2] by: t = sin pt) ds if t [ πp 0 1 s p ) 1/p 2, ] πp [ 2] πp sin p t) = sin p π p t) if t 2, 3πp 2 and is periodic on. Set also by definition cos p t) = d dt sin pt). The usual fundamental trigonometric identity can be generalized by sin p t) p + cos p t) p = 1, and so it is easily seen that cos p 1) p C 1 ). Clearly, if p = 2 one finds the usual quantities: π 2 = π, sin 2 = sin and cos 2 = cos. Theorem 4.3 [54, 74, 61]). Let, N [1, ) and D 0, ). Then the following hold 1) 1,p 1,p = λ, where λ1,p was defined in 4.3) and λ in 4.9). 2) For every fixed p 1, ), the map, N, D is continuous. 3) If > 0 then for every D 0, π N 1/] λ1,p,π / and equality holds if and only if D = π N 1/. If moreover N N, then,π / = λ1,p S N N 1/)), i.e. coincides with the first eigenvalue of the p-laplacian on the round sphere of,π / radius N 1/. 4) If = 0 then 0,N,D = p 1) π p ) p. D For 0 and p 2, it is not easy to give an explicit expression of the lower bound. At least one can give some lower bounds, for instance recently Li and Wang [48] obtained that 4.10) 1 p 1) p p-spectral gap for CD loc, N) spaces. N N 1 ) p/2 for > 0, p 2. Theorem 4.4. Let, d, m) be a metric measure space satisfying CD loc, N), for some, N with N 1, and assume moreover it is essentially non-branching. Let D 0, ) be the diameter of and fix p 1, ). Then for any Lipschitz function f L p, m) with f f p 2 mdx) = 0 it holds 4.11) fx) p mdx) f p x) mdx). In other terms it holds,d,m) λ1,p. Notice that for D = π N 1)/ and N N, it follows that,d,m) λ1,p S N N 1)/)). Proof. Since the space, d) is bounded, then the CD loc, N) condition implies that m) <. Noting that the inequality 4.11) is invariant under multiplication of m by a positive constant, we can assume without loss of generality that m) = 1. Observing that the function 4.12) f := f f p 2 Lip) verifies the hypothesis of Theorem 2.7, we can write = Y T with fx) = 0, m-a.e. y Y, m T = m q qdq), with m q = gq, ) h q L 1), where the density h q verifies 2.6) for q-a.e. q Q and 0 = fz) m q dz) = fgq, t)) h q t) L 1 dt) = fgq, t)) fgq, t)) p 2 h q t) L 1 dt) Dom gq, )) Q Dom gq, ))

16 16 FABIO CAVALLETTI AND ANDEA MONDINO for q-a.e. q Q. Now consider the map t f q t) := fgq, t)) and note that it is Lipschitz. Since diam Dom gq, ))) D, from the definition of F s and of λ1,p we deduce that f q t) p h q t) L 1 dt) f qt) p h q t) L 1 dt). Noticing that f qt) f gq, t)) one obtains that fx) p mdx) = fx) p mdx) T ) = fx) p m q dx) qdq) Q ) = f q t) p h q t) L 1 dt) qdq) Q Dom gq, )) ) f qt) p h q t) L 1 dt) qdq) Q Dom gq, )) f p x) gq, )) h q t) L 1) ) dx) qdq) Q = f p x) mdx), and the claim follows Almost rigidity for the p-spectral gap. Theorem 4.5 Almost equality in the p-spectral gap implies almost maximal diameter). Let N > 1, and p 1, ) be fixed. Then for every ε > 0 there exists δ = δε, N, p) such that the following holds. Let, d, m) be an essentially non-branching metric measure space satisfying CD N 1 δ, N + δ). If,d,m) λ1,p,n,π + δ, then diam ) π ε. Proof. As above, without loss of generality we can assume m) = 1. Assume by contradiction that there exists ε 0 > 0 such that for every δ > 0 we can find an essentially non-branching metric measure space, d, m) satisfying CD N 1 δ, N + δ), with m) = 1, such that diam ) π ε 0 but,d,m) λ1,p,n,π + δ. The very definition of,d,m) implies that there exists a function f Lip), with f f p 2 m = 0 and f p mdx) = 1, such that 4.13) f p x) mdx),d,m) + δ λ1,p,n,π + 2δ. On the other hand, Theorem 4.3 ensures that there exists η > 0 such that,n,d λ1,p,n,π + 2η, D [0, π ε 0]. Moreover, the continuity of, N, D guarantees that, for every D 0 0, 1) there exists δ 0 = δ 0 N, D 0 ) such that δ,n+δ,d λ1,p,n,d η δ [0, δ 0], D [D 0, 2π]. Since clearly by definition we have that λ1,p 0,N,D for every > 0, N 1, p 1, ), Theorem 4.3 gives that lim D 0 λ1,p δ,n+δ,d lim D 0 λ1,p 0,N+δ,D = + uniformly for δ [0, δ 0 N)]. The combination of the last two estimates yields 4.14) δ,n+δ,d λ1,p,n,π + η D [0, π ε 0], δ [0, δ 0 N)].

17 SHAP GEOMETIC AND FUNCTIONAL INEQUALITIES FO CD, N) SPACES 17 By repeating the proof of Theorem 4.4, and observing that by construction it holds diam Dom gq, )) π ε 0, we then obtain f p x) mdx) = f p x) gq, )) h q t) L 1) ) dx) qdq) Q Q Q ) f qt) p h q t) L 1 dt) qdq) Dom gq, )) δ,n+δ,diam Dom gq, )),N,π + η) =,N,π + η. Contradicting 4.13), once chosen δ < η/2. Q f q t) p h q t) L 1 dt) Dom gq, )) ) f q t) p h q t) L 1 dt) Dom gq, )) Corollary 4.6 Almost equality in the p-spectral gap implies mgh-closeness to a spherical suspension). Let N 2, and p 1, ) be fixed. Then for every ε > 0 there exists δ = δε, N, p) > 0 such that the following holds. Let, d, m) be an CD N 1 δ, N + δ) space. If,d,m) λ1,p,n,π + δ, then there exists an CD N 2, N 1) space Y, d Y, m Y ) such that d mgh, d, m), [0, π] sin Y ) ε. Proof. Fix N [2, ), p 1, ) and assume by contradiction there exist ε 0 > 0 and a sequence j, d j, m j ) of CD N 1 1 j, N + 1 j ) spaces such that λ1,p,d,m) λ1,p,n,π + 1 j, but 4.15) d mgh j, [0, π] sin Y ) ε 0 for every j N qdq) ) qdq) and every CD N 2, N 1) space Y, d Y, m Y ) with m Y Y ) = 1. Observe that Theorem 4.5 yields 4.16) diam j, d j )) π. By the compactness/stability property of CD, N) spaces recalled in Theorem 2.6 we get that, up to subsequences, the spaces j mgh-converge to a limit CD N 1, N) space, d, m ). Since the diameter is continuous under mgh convergence of uniformly bounded spaces, 4.16) implies that diam, d )) = π. But then by the Maximal Diameter Theorem [43] we get that, d, m ) is isomorphic to a spherical suspension [0, π] sin Y for some CD N 2, N 1) space Y, d Y, m Y ) with m Y Y ) = 1. Clearly this contradicts 4.15) and the thesis follows. Corollary 4.7 p-obata Theorem). Let, d, m) be an CD N 1, N) space for some N 2, and let 1 < p <. If,d,m) = λ1,p,n,π = S N ), if N N), then, d, m) is a spherical suspension, i.e. there exists an CD N 2, N 1) space Y, d Y, m Y ) such that, d, m) is isomorphic to [0, π] sin Y. Proof. Theorem 4.5 implies that diam, d)) = π and the thesis then follows by the Maximal Diameter Theorem [43]. emark 4.8. The Obata s Theorem for p = 2 in CD N 1, N) spaces has been recently obtained by etterer [43] by different methods see also [41]); the approach proposed here has the double advantage of length and of being valid for every p 1, ).

18 18 FABIO CAVALLETTI AND ANDEA MONDINO 5. The case p = 1 and the Cheeger constant It is well known see for instance [40, 76]) that an alternative way of defining,d,m) which extends also to p = 1 is the following. For every p [1, ) and every f L p ) let 1/p c p f) := inf f c m) p. c For every p 1, ) it holds that [40, Corollary 2.11] { },d,m) = inf f p m : f Lip L p ), c p f) = f L p = 1. It is then natural to set { 5.1) λ 1,1,d,m) = inf } f m : f Lip L 1 ), c 1 f) = f L 1 = 1. Assuming that m) = 1, recall that a number M f is a median for f if and only if m{f M f }) 1 2 and m{f M f }) 1 2. It is not difficult to check that see for instance [19, Section VI]) for every f L 1 ) there exists a median of f, and moreover f M f m = c 1 f) holds for every median M f of f. This link between c 1 f) and M f is useful to prove the equivalence between the Cheeger constant and λ 1,1,d,m). ecall that the Cheeger constant h,d,m) is defined by { m + } E) h,d,m) := inf : E is Borel and me) 0, 1/2], me) where m + me ε ) me) E) := lim inf ε 0 ε is the outer) Minkowski content. As usual E ε := {x : y E such that dx, y) < ε} is the ε-neighborhood of E with respect to the metric d. The next result, due to Maz ya [55] and Federer- Fleming [29] see also [10] for a careful derivation, [56, Lemma 2.2] and [40, Proposition 2.13] for the present formulation), rewrites Cheeger s isoperimetric inequality in functional form. Proposition 5.1. Assume that, d, m) is a m.m.s with m{x}) = 0 for every x, i.e. m is atomless. Then h,d,m) = λ 1,1,d,m). It is then clear that the comparison and almost rigidity theorems for λ 1,1 will be based on the corresponding isoperimetric ones obtained by the authors in [17]. To this aim in the next subsection we briefly recall the model Cheeger constant for the comparison The model Cheeger constant h. If > 0 and N N, by the Lévy-Gromov isoperimetric inequality we know that, for N-dimensional smooth manifolds having icci curvature bounded below by, the Cheeger constant i is bounded below by the one of the N-dimensional round sphere of the suitable radius. In other words the model Cheeger constant is the one of S N. For N 1, arbitrary real numbers the situation is more complicated, and just recently E. Milman [57] discovered what is the model Cheeger constant more precisely he discovered the model isoperimetric profile, which in turn implies the model Cheeger constant). In this short section we recall its definition. Given δ > 0, set sin δt)/ δ δ > 0 s δ t) := t δ = 0 sinh δt)/ δ δ < 0 cos δt) δ > 0, c δ t) := 1 δ = 0 cosh δt) δ < 0.