Spectral gap for some invariant log-concave probability measures

We show that the conjecture of Kannan, Lov\'{a}sz, and Simonovits on isoperimetric properties of convex bodies and log-concave measures, is true for log-concave measures of the form $\rho(|x|_B)dx$ on $\mathbb{R}^n$ and $\rho(t,|x|_B) dx$ on $\mathbb{R}^{1+n}$, where $|x|_B$ is the norm associated to any convex body $B$ already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.Comment: To appear in Mathematika. This version can differ from the one published in Mathematik

Approaching Eldan’s and Lee & Vempala’s bounds for the KLS conjecture in a unified method

La principal idea de este artículo es revisar las pruebas de las mejores estimaciones conocidas para la conjetura KLS de salto espectral, demostradas por Eldan y Lee & Vempala, aplicando el esquema de localización de Eldan a dos sistemas de ecuaciones diferenciales estocásticas diferentes. Damos una prueba unificada de estas dos acotaciones obteniendo la estimación de Eldan desde el sistema de ecuaciones diferenciales estocásticas considerado por Lee & Vempala

Distribution of mass in high-dimensional convex bodies

In this paper we will explore the interaction between convex geometry and proba-bility in the study of the distribution of volume in high-dimensional convex bodies. On the one hand, a convex body K in Rn can be understood as a probability space whenthe normalized Lebesgue measure is considered. Thus, probabilistic tools become veryhandy in the study of the behavior of a random vector uniformly distributed inK.This leads to the understanding of how the volume is distributed in a convex body andthe obtention of geometric inequalities. On the other hand, when considering lower-dimensional marginals of the uniform probability measure onK, we leave the class ofuniform probabilities on convex bodies but remain in the class of log-concave probabilities. Many geometric inequalities can be extended to the context of log-concaveprobabilities, leading to functional inequalities for log- concave functions

Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation

We give the first rigorous proof of the convergence of Riemannian Hamiltonian Monte Carlo, a general (and practical) method for sampling Gibbs distributions. Our analysis shows that the rate of convergence is bounded in terms of natural smoothness parameters of an associated Riemannian manifold. We then apply the method with the manifold defined by the log barrier function to the problems of (1) uniformly sampling a polytope and (2) computing its volume, the latter by extending Gaussian cooling to the manifold setting. In both cases, the total number of steps needed is O^{*}(mn^{\frac{2}{3}}), improving the state of the art. A key ingredient of our analysis is a proof of an analog of the KLS conjecture for Gibbs distributions over manifolds