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Emanuel Milman

Publications and source records attributed to Emanuel Milman.

At least 37 records · Page 2Linked to original sources
arXiv2021

The Gaussian Double-Bubble and Multi-Bubble Conjectures

We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into $q$ cells of prescribed (positive) Gaussian measure when $2 \leq q \leq n+1$, is to use a "simplicial cluster", obtained from the Voronoi cells of $q$ equidistant points. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers (up to null-sets). In particular, the case $q=3$ confirms the Gaussian Double-Bubble Conjecture: the unique least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ ($n \geq 2$) into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an $(n-2)$-dimensional plane at $120^{\circ}$ angles (forming a tripod or "Y" shape in the plane). The case $q=2$ recovers the classical Gaussian isoperimetric inequality. To establish the Multi-Bubble conjecture, we show that in the above range of $q$, stable regular clusters must have flat interfaces, therefore consisting of convex polyhedral cells (with at most $q-1$ facets). In the Double-Bubble case $q=3$, it is possible to avoid establishing flatness of the interfaces by invoking a certain dichotomy on the structure of stable clusters, yielding a simplified argument.

math.DG↗
arXiv2021

The Gaussian Double-Bubble Conjecture

We establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an $(n-2)$-dimensional plane at $120^{\circ}$ angles (forming a tripod or "Y" shape in the plane). Moreover, we prove that tripod-clusters are the unique isoperimetric minimizers (up to null-sets).

math.FA↗
arXiv2021

Regular Random Sections of Convex Bodies and the Random Quotient-of-Subspace Theorem

It was shown by G. Pisier that any finite-dimensional normed space admits an $α$-regular $M$-position, guaranteeing not only regular entropy estimates but moreover regular estimates on the diameters of minimal sections of its unit-ball and its dual. We revisit Pisier's argument and show the existence of a \emph{different} position, which guarantees the same estimates for \emph{randomly sampled} sections \emph{with high-probability}. As an application, we obtain a \emph{random} version of V. Milman's Quotient-of-Subspace Theorem, asserting that in the above position, \emph{typical} quotients of subspaces are isomorphic to Euclidean, with a distance estimate which matches the best-known deterministic one (and beating all prior estimates which hold with high-probability). Our main novel ingredient is a new position of convex bodies, whose existence we establish by using topological arguments and a fixed-point theorem.

math.FA↗
arXiv2021

The Globalization Theorem for the Curvature Dimension Condition

The Lott-Sturm-Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space $(X,{\mathsf d},{\mathfrak m})$ (so that $(\text{supp} \; {\mathfrak m},{\mathsf d})$ is a length-space and ${\mathfrak m}(X) < \infty$) verifying the local Curvature-Dimension condition $\mathsf{CD}_{loc}(K,N)$ with parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$, also verifies the global Curvature-Dimension condition $\mathsf{CD}(K,N)$. In other words, the Curvature-Dimension condition enjoys the globalization (or local-to-global) property, answering a question which had remained open since the beginning of the theory. For the proof, we establish an equivalence between $L^1$ and $L^2$ optimal-transport-based interpolation. The challenge is not merely a technical one, and several new conceptual ingredients which are of independent interest are developed: an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order temporal derivative of associated Kantorovich potentials; a surprising third-order theory for the latter Kantorovich potentials, which holds in complete generality on any proper geodesic space; and a certain rigidity property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. As a consequence, numerous variants of the Curvature-Dimension condition proposed by various authors throughout the years are shown to, in fact, all be equivalent in the above setting, thereby unifying the theory.

math.MG↗
arXiv2020

Reverse Hölder Inequalities for log-Lipschitz Functions

Reverse Hölder inequalities for a class of functions on a probability space constitute an important tool in Analysis in Probability. After revisiting how a (modified) log-Sobolev inequality can be used to derive reverse Hölder inequalities for the class of log-Lipschitz functions, we obtain a weaker condition using general Transport-Entropy inequalities, which can also handle approximately log-Lipschitz functions. In its weakest form, the condition degenerates to the assumption of satisfying a concentration inequality. We compare this with a scenario in which the underlying space only satisfies a Poincaré inequality.

math.FA↗
arXiv2020

The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifolds

We obtain the best known quantitative estimates for the $L^p$-Poincaré and log-Sobolev inequalities on domains in various sub-Riemannian manifolds, including ideal Carnot groups and in particular ideal generalized H-type Carnot groups and the Heisenberg groups, corank $1$ Carnot groups, the Grushin plane, and various H-type foliations, Sasakian and $3$-Sasakian manifolds. Moreover, this constitutes the first time that a quantitative estimate independent of the dimension is established on these spaces. For instance, the Li-Yau / Zhong-Yang spectral-gap estimate holds on all Heisenberg groups of arbitrary dimension up to a factor of $4$. We achieve this by introducing a quasi-convex relaxation of the Lott-Sturm-Villani $\mathsf{CD}(K,N)$ condition we call the Quasi Curvature-Dimension condition $\mathsf{QCD}(Q,K,N)$. Our motivation stems from a recent interpolation inequality along Wasserstein geodesics in the ideal sub-Riemannian setting due to Barilari and Rizzi. We show that on an ideal sub-Riemannian manifold of dimension $n$, the Measure Contraction Property $\mathsf{MCP}(K,N)$ implies $\mathsf{QCD}(Q,K,N)$ with $Q = 2^{N-n} \geq 1$, thereby verifying the latter property on the aforementioned ideal spaces; a result of Balogh-Kristály-Sipos is used instead to handle non-ideal corank $1$ Carnot groups. By extending the localization paradigm to completely general interpolation inequalities, we reduce the study of various analytic and geometric inequalities on $\mathsf{QCD}$ spaces to the one-dimensional case. Consequently, we deduce that while (strictly) sub-Riemannian manifolds do not satisfy any type of $\mathsf{CD}$ condition, many of them satisfy numerous functional inequalities with \emph{exactly the same} quantitative dependence (up to a factor of $Q$) as their $\mathsf{CD}$ counterparts.

math.FA↗
arXiv2020

Sharp Poincaré inequalities under Measure Contraction Property

We prove a sharp Poincaré inequality for subsets $Ω$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\textrm{MCP}(K,N)$, whose diameter is bounded above by $D$. This is achieved by identifying the corresponding one-dimensional model densities and a localization argument, ensuring that the Poincaré constant we obtain is best possible as a function of $K$, $N$ and $D$. Another new feature of our work is that we do not need to assume that $Ω$ is geodesically convex, by employing the geodesic hull of $Ω$ on the energy side of the Poincaré inequality. In particular, our results apply to geodesic balls in ideal sub-Riemannian manifolds, such as the Heisenberg group.

math.MG↗
arXiv2018

Reciprocals and Flowers in Convexity

We study new classes of convex bodies and star bodies with unusual properties. First we define the class of reciprocal bodies, which may be viewed as convex bodies of the form "$1/K$". The map $K\mapsto K^\prime$ sending a body to its reciprocal is a duality on the class of reciprocal bodies, and we study its properties. To connect this new map with the classic polarity we use another construction, associating to each convex body $K$ a star body which we call its flower and denote by $K^\clubsuit$. The mapping $K\mapsto K^\clubsuit$ is a bijection between the class $\mathcal{K}_0^n$ of convex bodies and the class $\mathcal{F}^n$ of flowers. We show that the polarity map $\circ:\mathcal{K}_0^n\to\mathcal{K}_0^n$ decomposes into two separate bijections: First our flower map $\clubsuit:\mathcal{K}_0^n\to\mathcal{F}^n$, followed by the spherical inversion $Φ$ which maps $\mathcal{F}^n$ back to $\mathcal{K}_0^n$. Each of these maps has its own properties, which combine to create the various properties of the polarity map. We study the various relations between the four maps $\prime$, $\circ$, $\clubsuit$ and $Φ$ and use these relations to derive some of their properties. For example, we show that a convex body $K$ is a reciprocal body if and only if its flower $K^\clubsuit$ is convex. We show that the class $\mathcal{F}^n$ has a very rich structure, and is closed under many operations, including the Minkowski addition. This structure has corollaries for the other maps which we study. For example, we show that if $K$ and $T$ are reciprocal bodies so is their "harmonic sum" $(K^\circ+T^\circ)^\circ$. We also show that the volume $\left|\left(\sum_iλ_{i}K_i\right)^\clubsuit\right|$ is a homogeneous polynomial in the $λ_i$'s, whose coefficients can be called "$\clubsuit$-type mixed volumes". Related geometric inequalities are also derived.

math.MG↗
arXiv2018

Generalized Dual Sudakov Minoration via Dimension Reduction - A Program

We propose a program for establishing a conjectural extension to the class of (origin-symmetric) log-concave probability measures $μ$, of the classical dual Sudakov Minoration on the expectation of the supremum of a Gaussian process: \begin{equation} \label{eq:abstract} M(Z_p(μ), C \int ||x||_K dμ\cdot K) \leq \exp(C p) \;\;\, \forall p \geq 1 . \end{equation} Here $K$ is an origin-symmetric convex body, $Z_p(μ)$ is the $L_p$-centroid body associated to $μ$, $M(A,B)$ is the packing-number of $B$ in $A$, and $C > 0$ is a universal constant. The Program consists of first establishing a Weak Generalized Dual Sudakov Minoration, involving the dimension $n$ of the ambient space, which is then self-improved to a dimension-free estimate after applying a dimension-reduction step. The latter step may be thought of as a conjectural "small-ball one-sided" variant of the Johnson--Lindenstrauss dimension-reduction lemma. We establish the Weak Generalized Dual Sudakov Minoration for a variety of log-concave probability measures and convex bodies (for instance, this step is fully resolved assuming a positive answer to the Slicing Problem). The Separation Dimension-Reduction step is fully established for ellipsoids and, up to logarithmic factors in the dimension, for cubes, resulting in a corresponding Generalized (regular) Dual Sudakov Minoration estimate for these bodies and arbitrary log-concave measures, which are shown to be (essentially) best-possible. Along the way, we establish a regular version of (\ref{eq:abstract}) for all $p \geq n$ and provide a new direct proof of Sudakov Minoration via The Program.

math.FA↗
arXiv2018

Spectral Estimates, Contractions and Hypercontractivity

Sharp comparison theorems are derived for all eigenvalues of the (weighted) Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed with strongly log-concave and log-convex densities, extensions to $p$-exponential measures, unit-balls of $\ell_p^n$, one-dimensional spaces and Riemannian submersions. Our main tool is a general Contraction Principle for "eigenvalues" on arbitrary metric-measure spaces. Motivated by Caffarelli's Contraction Theorem, we put forth several conjectures pertaining to the existence of contractions from the canonical sphere (and Gaussian space) to weighted-manifolds of appropriate topological type having (generalized) Ricci curvature positively bounded below; these conjectures are consistent with all known isoperimetric, heat-kernel and Sobolev-type properties, and would imply sharp conjectural spectral estimates on such spaces. While we do not resolve these conjectures for the individual eigenvalues, we verify their Weyl asymptotic distribution in the compact and non-compact settings, obtain non-asymptotic estimates using the Cwikel--Lieb--Rozenblum inequality, and estimate the trace of the associated heat-kernel assuming that the associated heat semi-group is hypercontractive. As a side note, an interesting trichotomy for the heat-kernel is obtained.

math.SP↗
arXiv2018

Local $L^p$-Brunn-Minkowski inequalities for $p < 1$

The $L^p$-Brunn-Minkowski theory for $p\geq 1$, proposed by Firey and developed by Lutwak in the 90's, replaces the Minkowski addition of convex sets by its $L^p$ counterpart, in which the support functions are added in $L^p$-norm. Recently, Böröczky, Lutwak, Yang and Zhang have proposed to extend this theory further to encompass the range $p \in [0,1)$. In particular, they conjectured an $L^p$-Brunn-Minkowski inequality for origin-symmetric convex bodies in that range, which constitutes a strengthening of the classical Brunn-Minkowski inequality. Our main result confirms this conjecture locally for all (smooth) origin-symmetric convex bodies in $\mathbb{R}^n$ and $p \in [1 - \frac{c}{n^{3/2}},1)$. In addition, we confirm the local log-Brunn--Minkowski conjecture (the case $p=0$) for small-enough $C^2$-perturbations of the unit-ball of $\ell_q^n$ for $q \geq 2$, when the dimension $n$ is sufficiently large, as well as for the cube, which we show is the conjectural extremal case. For unit-balls of $\ell_q^n$ with $q \in [1,2)$, we confirm an analogous result for $p=c \in (0,1)$, a universal constant. It turns out that the local version of these conjectures is equivalent to a minimization problem for a spectral-gap parameter associated with a certain differential operator, introduced by Hilbert (under different normalization) in his proof of the Brunn-Minkowski inequality. As applications, we obtain local uniqueness results in the even $L^p$-Minkowski problem, as well as improved stability estimates in the Brunn-Minkowski and anisotropic isoperimetric inequalities.

math.FA↗
arXiv2018

The KLS Isoperimetric Conjecture for Generalized Orlicz Balls

What is the optimal way to cut a convex bounded domain $K$ in Euclidean space $(\mathbb{R}^n,|\cdot|)$ into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovász and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of $n$) in the surface area, one might as well dissect $K$ using a hyperplane. This conjectured essential equivalence between the former non-linear isoperimetric inequality and its latter linear relaxation, has been shown over the last two decades to be of fundamental importance to the understanding of volumetric and spectral properties of convex domains. In this work, we address the conjecture for the subclass of generalized Orlicz balls \[ K = \left \{x \in \mathbb{R}^n \; ; \; \sum_{i=1}^n V_i(x_i) \leq E \right \} , \] confirming its validity for certain levels $E \in \mathbb{R}$ under a mild technical assumption on the growth of the convex functions $V_i$ at infinity (without which we confirm the conjecture up to a $\log(1+n)$ factor). In sharp contrast to previous approaches for tackling the KLS conjecture, we emphasize that no symmetry is required from $K$. This significantly enlarges the subclass of convex bodies for which the conjecture is confirmed.

math.FA↗
arXiv2017

Brascamp-Lieb type inequalities on weighted Riemannian manifolds with boundary

It is known that by dualizing the Bochner-Lichnerowicz-Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Brascamp-Lieb type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp-Lieb, Bobkov-Ledoux and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Our framework allows to encompass the entire class of Borell's convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative generalized dimension.

math.DG↗
arXiv2017

Poincaré and Brunn--Minkowski inequalities on the boundary of weighted Riemannian manifolds

We study a Riemannian manifold equipped with a density which satisfies the Bakry--Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is locally-convex; this generalizes a purely Euclidean inequality of Colesanti, originally derived as an infinitesimal form of the Brunn-Minkowski inequality, thereby precluding any extensions beyond the Euclidean setting. A dual version for generalized mean-convex boundaries is also obtained, yielding spectral-gap estimates for the weighted Laplacian on the boundary. Motivated by these inequalities, a new geometric evolution equation is proposed, which extends to the Riemannian setting the Minkowski addition operation of convex domains, a notion thus far confined to the purely linear setting. This geometric flow is characterized by having parallel normals (of varying velocity) to the evolving hypersurface along the trajectory, and is intimately related to a homogeneous Monge-Ampère equation on the exterior of the convex domain. Using the aforementioned Poincaré-type inequality on the boundary of the evolving hypersurface, we obtain a novel Brunn--Minkowski inequality in the weighted-Riemannian setting, amounting to a certain concavity property for the weighted-volume of the evolving enclosed domain. All of these results appear to be new even in the classical non-weighted Riemannian setting.

math.DG↗
arXiv2016

Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the range $N \in (-\infty,1)$, extending the scope from the traditional range $N \in [n,\infty]$. In particular, we identify the correct one-dimensional model-spaces under an additional diameter upper bound, and discover a new case yielding a \emph{single} model space (besides the previously known $N$-sphere and Gaussian measure when $N \in [n,\infty]$): a (positively curved) sphere of (possibly negative) dimension $N \in (-\infty,1)$. When curvature is non-negative, we show that arbitrarily weak concentration implies an $N$-dimensional Cheeger isoperimetric inequality, and derive various weak Sobolev and Nash-type inequalities on such spaces. When curvature is strictly positive, we observe that such spaces satisfy a Poincaré inequality uniformly for all $N \in (-\infty,1-ε]$, and enjoy a two-level concentration of the type $\exp(-\min(t,t^2))$. Our main technical tool is a generalized version of the Heintze--Karcher theorem, which we extend to the range $N \in (-\infty,1)$.

math.DG↗
arXiv2016

Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities

Given a probability measure $μ$ supported on a convex subset $Ω$ of Euclidean space $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincaré and log-Sobolev type inequalities on $(Ω,g_0,μ)$. To this end, we change the metric $g_0$ to a more general Riemannian one $g$, adapted in a certain sense to $μ$, and perform our analysis on $(Ω,g,μ)$. The types of metrics we consider are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when $μ$ is unconditional, i.e. invariant under reflection with respect to the principle hyperplanes), and metrics conformal to the Euclidean one, which have not been previously explored in this context. Invoking on $(Ω,g,μ)$ tools such as Riemannian generalizations of the Brascamp--Lieb inequality and the Bakry--Émery criterion, and passing back to the original Euclidean metric, we obtain various weighted inequalities on $(Ω,g_0,μ)$: refined and entropic versions of the Brascamp--Lieb inequality, weighted Poincaré and log-Sobolev inequalities, Hardy-type inequalities, etc. Key to our analysis is the positivity of the associated Lichnerowicz--Bakry--Émery generalized Ricci curvature tensor, and the convexity of the manifold $(Ω,g,μ)$. In some cases, we can only ensure that the latter manifold is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.

math.FA↗
arXiv2016

$M$-estimates for isotropic convex bodies and their $L_q$-centroid bodies

Let $K$ be a centrally-symmetric convex body in $\mathbb{R}^n$ and let $\|\cdot\|$ be its induced norm on ${\mathbb R}^n$. We show that if $K \supseteq r B_2^n$ then: \[ \sqrt{n} M(K) \leqslant C \sum_{k=1}^{n} \frac{1}{\sqrt{k}} \min\left(\frac{1}{r} , \frac{n}{k} \log\Big(e + \frac{n}{k}\Big) \frac{1}{v_{k}^{-}(K)}\right) . \] where $M(K)=\int_{S^{n-1}} \|x\|\, dσ(x)$ is the mean-norm, $C>0$ is a universal constant, and $v^{-}_k(K)$ denotes the minimal volume-radius of a $k$-dimensional orthogonal projection of $K$. We apply this result to the study of the mean-norm of an isotropic convex body $K$ in ${\mathbb R}^n$ and its $L_q$-centroid bodies. In particular, we show that if $K$ has isotropic constant $L_K$ then: \[ M(K) \leqslant \frac{C\log^{2/5}(e+ n)}{\sqrt[10]{n}L_K} . \]

math.FA↗