Estimating the average of functions with convexity properties by means of a new center

TL;DR

This paper characterizes the sets and concave functions that maximize a convex integral functional over convex sets with fixed volume, extending previous results and providing new estimates on convex set sections.

Contribution

It introduces a new point-based condition to identify extremal functions and sets for a class of convex integral optimization problems, extending prior theoretical results.

Findings

Characterization of extremal sets and functions for the supremum of a convex integral functional.

Extension of Milman and Pajor's results to broader classes of convex functions.

New estimates on the volume of convex set sections passing through specific points.

Abstract

In this article we show the following result: if $C$ is an $n$-dimensional convex and compact subset, $f:C\rightarrow[0,\infty)$ is concave, and $\phi:[0,\infty)\rightarrow[0,\infty)$ is a convex function with $\phi(0)=0$, we then characterize the class of sets and concave functions that attain the supremum \[ \sup_{C,f}\int_C\phi(f(x))dx, \] where the supremum ranges over all sets $C$ with $n$-dimensional volume $|C|=c$ and the additional condition that $f(x_{C,f})=k$ for some point $x_{C,f}\in C$ that we introduce in the article, for two non-negative constants $c,k>0$. As a consequence, we extend some results of Milman and Pajor in [MP] and some in [Thm. 1.2, GoMe]. Besides, we also obtain some new estimates on the volume of particular sections of a convex set $K$ passing through a new point of $K$.