On a generalization of the Hermite-Hadamard inequality and applications in convex geometry

TL;DR

This paper generalizes the Hermite-Hadamard inequality for convex geometry, providing new bounds for convex sets and log-concave functions, with applications in volume estimation and geometric analysis.

Contribution

It introduces a novel inequality extending Hermite-Hadamard to convex sets and functions, with conditions for equality and applications in volume estimation.

Findings

Established a new inequality for convex sets and concave functions.

Derived volume estimates for convex sets using central sections.

Extended results to log-concave functions.

Abstract

In this paper we show the following result: if C is an n-dimensional 0-symmetric convex compact set, $f:C\rightarrow[0,1)$ is concave, and $g:[0,1)\rightarrow[0,1)$ is not identically zero, convex, with g(0)=0, then \[ \frac{1}{|C|}\int_C g(f(x))dx \leq \frac12 \int_{-1}^1g(f(0)(1+t))dt, \] where |C| denotes the volume of C. If g? is strictly convex, equality holds if and only if f is affine, C is a generalized symmetric cylinder and f becomes 0 at one of the basis of C. We exploit this inequality to answer a question of Francisco Santos on estimating the volume of a convex set by means of the volume of a central section of it. Second, we also derive a corresponding estimate for log-concave functions.