The Busemann-Petty problem on entropy of log-concave functions

TL;DR

This paper explores an entropy version of the Busemann-Petty problem for log-concave functions, establishing positive results in low dimensions and confirming the negative case in higher dimensions.

Contribution

It extends the classical Busemann-Petty problem to the entropy of log-concave functions, providing new insights and results for dimensions 2 to 4 and confirming the negative case for dimensions 5 and above.

Findings

Positive answer for dimensions 2 to 4.

Negative answer for dimensions 5 and above.

Includes the classical Busemann-Petty problem as a special case.

Abstract

The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space $\mathbb{R}^n$ with smaller central hyperplane sections necessarily have smaller volume. The solution has been completed and the answer is affirmative if $n \le 4$ and negative if $n\ge 5$. In this paper, we investigate the Busemann-Petty problem on entropy of log-concave functions: For even log-concave functions $f$ and $g$ with finite positive integrals in $\mathbb{R}^n$, if the marginal $\int_{\mathbb{R}^n\cap H}f(x)dx$ of $f$ is smaller than the marginal $\int_{\mathbb{R}^n\cap H}g(x)dx$ of $g$ for every hyperplane $H$ passing through the origin, whether the entropy ${\rm Ent}(f)$ of $f$ is bigger than the entropy ${\rm Ent}(g)$ of $g$? The Busemann-Petty problem on entropy of log-concave functions includes the Busemann-Petty problem, hence, its answer is negative when $n\geq5$. For $2\leq n\leq4$ we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions.