%0 Journal Article
%T The entropic barrier: a simple and optimal universal self-concordant barrier
%A S¨¦bastien Bubeck
%A Ronen Eldan
%J Mathematics
%D 2014
%I arXiv
%X We prove that the Cram\'er transform of the uniform measure on a convex body in $\mathbb{R}^n$ is a $(1+o(1)) n$-self-concordant barrier, improving a seminal result of Nesterov and Nemirovski. This gives the first explicit construction of a universal barrier for convex bodies with optimal self-concordance parameter. The proof is based on basic geometry of log-concave distributions, and elementary duality in exponential families.
%U http://arxiv.org/abs/1412.1587v3