Isoperimetric inequality under Measure-Contraction property

Abstract

We prove that if (X, d, m) is an essentially non-branching metric measure space with m(X) = 1, having Ricci curvature bounded from below by K and dimension bounded above by N ∈ (1, ∞), understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequalityà la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained. model isoperimetric profile I CD K,N,D such that if a Riemannian manifold with density verifying CD(K, N ) has diameter at most D > 0, then the isoperimetric profile function of the weighted manifold is bounded from below by I CD K,N,D . After the works of Cordero-Erausquin-McCann-Schmuckenshläger [25], Otto-Villani [42] and von Renesse-Sturm [47], it was realized that the CD(K, ∞) condition in the smooth setting may be equivalently formulated synthetically as a certain convexity property of an entropy functional along W 2 Wasserstein geodesics (associated to L 2 -Optimal-Transport). This idea led Lott-Villani [36] and Sturm [51, 52], to propose a successful (and compatible with the classical one) synthetic definition of CD(K, N ) for a general (complete, separable) metric space (X, d) endowed with a (locally-finite Borel) reference measure m ("metricmeasure space", or m.m.s.); the theory of m.m.s.'s verifying CD(K, N ) has then extensively developed leading to a rich and fruitful approach to the geometry of m.m.s.'s by means of 3,4,27,26,5,38,29,33,13]. See also [1] for a recent account on the topic. Building on the work by Klartag [34] and the localization paradigm developed by Payne-Weinberger [44], Gromov-Milman [31] and Kannan-Lovász-Simonovits [32], the first author with Mondino [19] managed to extend Lévy-Gromov-Milman isoperimetric inequality to the class of essentially non-branching (see Section 2 for the definition) m.m.s.'s verifying CD(K, N ) with m(X) = 1; in particular [19] proves that

Isoperimetric inequality in noncompact MCP spaces

“…As one can expect from the sharpness of the isoperimetric inequality for compact MCP(K, N ) spaces obtained in [12], also inequality (3.5) is sharp. In particular, if we fix a, v > 0, N > 1, we can find a MCP(0, N ) space (X, d, m), with AVR X = a, and a subset E ⊂ X such that m + (E) = (N ω N AVR X )…”

“…A sharp isoperimetric inequality à la Levy-Gromov for the class of spaces verifying MCP(K, N ) with finite diameter and total mass 1 is the main content of [12]. The sharp lower bound on the isoperimetric profile function (2.2) has the following representation:…”

Sharp p-Poincaré inequalities under measure contraction property

“…Remark 3.7. The difference between the cases K ≤ 0 and K > 0 was already observed in [13] in the isoperimetric context and in [17] in the 2-Poincaré context. It is known that the monotonicity property (3.9) is false when K > 0.…”

“…It can be seen that (c.f. Lemma 3.4 [13]) h K,N,D does not satisfy any forms of CD condition. Theorem 3.10 (One dimensional p-spectral gap).…”

The Quasi Curvature‐Dimension Condition with Applications to Sub‐Riemannian Manifolds

“…In the past year, the study of MCP spaces has seen some increased activity, starting from the work of Cavalletti and Santarcangelo [33], who obtained sharp isoperimetric inequalities, and continuing with the work of Han-Milman [46] and Han [45], who obtained sharp Poincaré and L p -Poincaré inequalities, respectively, for MCP.K; N / spaces whose diameter is upper-bounded by D P .0; I/. While these results are sharp for the class of MCP spaces, as witnessed by equipping .R; j ¡ j/ with an appropriate measure m, it remained unclear whether they provide good quantitative estimates for the above specific examples from the sub-Riemmanian setting, which certainly have more structure than general MCP spaces.…”