The entropic barrier is $n$-self-concordant

TL;DR

This paper demonstrates that the entropic barrier for convex bodies in n-dimensional space is optimally n-self-concordant, leveraging the dimensional Brascamp-Lieb inequality to establish this bound.

Contribution

It shows that the optimal n-bound on the self-concordance parameter of the entropic barrier follows from the dimensional Brascamp-Lieb inequality, confirming the conjectured bound.

Findings

The entropic barrier is n-self-concordant.

The optimal bound of n on the self-concordance parameter is proven.

The proof relies on the dimensional Brascamp-Lieb inequality.

Abstract

For any convex body $K \subseteq \mathbb R^n$, S. Bubeck and R. Eldan introduced the entropic barrier on $K$ and showed that it is a $(1+o(1)) \, n$-self-concordant barrier. In this note, we observe that the optimal bound of $n$ on the self-concordance parameter holds as a consequence of the dimensional Brascamp-Lieb inequality.