Optimal Concentration of Information Content For Log-Concave Densities

An elementary proof is provided of sharp bounds for the varentropy of random vectors with log-concave densities, as well as for deviations of the information content from its mean. These bounds significantly improve on the bounds obtained by Bobkov and Madiman ({\it Ann. Probab.}, 39(4):1528--1543, 2011).Comment: 15 pages. Changes in v2: Remark 2.5 (due to C. Saroglou) added with more general sufficient conditions for equality in Theorem 2.3. Also some minor corrections and added reference

On the mean width of log-concave functions

In this work we present a new, natural, definition for the mean width of log-concave functions. We show that the new definition coincide with a previous one by B. Klartag and V. Milman, and deduce some properties of the mean width, including an Urysohn type inequality. Finally, we prove a functional version of the finite volume ratio estimate and the low-M* estimate.Comment: 15 page

Remarks on the KLS conjecture and Hardy-type inequalities

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $\Omega \subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial \Omega$. This reduces the study of the Neumann Poincar\'e constant on $\Omega$ to that of the cone and Lebesgue measures on $\partial \Omega$; these may be bounded via the curvature of $\partial \Omega$. A second reduction is obtained to the class of harmonic functions on $\Omega$. We also study the relation between the Poincar\'e constant of a log-concave measure $\mu$ and its associated K. Ball body $K_\mu$. In particular, we obtain a simple proof of a conjecture of Kannan--Lov\'asz--Simonovits for unit-balls of $\ell^n_p$, originally due to Sodin and Lata{\l}a--Wojtaszczyk.Comment: 18 pages. Numbering of propositions, theorems, etc.. as appeared in final form in GAFA seminar note

Estimates for measures of sections of convex bodies

A $\sqrt{n}$ estimate in the hyperplane problem with arbitrary measures has recently been proved in \cite{K3}. In this note we present analogs of this result for sections of lower dimensions and in the complex case. We deduce these inequalities from stability in comparison problems for different generalizations of intersection bodies

Estimation in high dimensions: a geometric perspective

This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of asymptotic convex geometry, and demonstrates connections between geometric results and estimation problems. The theory is illustrated with applications to sparse recovery, matrix completion, quantization, linear and logistic regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change

Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems

We obtain new principles for transferring log-Sobolev and Spectral-Gap inequalities from a source metric-measure space to a target one, when the curvature of the target space is bounded from below. As our main application, we obtain explicit estimates for the log-Sobolev and Spectral-Gap constants of various conservative spin system models, consisting of non-interacting and weakly-interacting particles, constrained to conserve the mean-spin. When the self-interaction is a perturbation of a strongly convex potential, this partially recovers and partially extends previous results of Caputo, Chafa\"{\i}, Grunewald, Landim, Lu, Menz, Otto, Panizo, Villani, Westdickenberg and Yau. When the self-interaction is only assumed to be (non-strongly) convex, as in the case of the two-sided exponential measure, we obtain sharp estimates on the system's spectral-gap as a function of the mean-spin, independently of the size of the system.Comment: 57 page