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

Bernardo González Merino

Published 2020-05-28Version 1

In this article we show the following result: if $C$ is an n-dimensional convex and compact set, $f:C\rightarrow[0,\infty)$ is concave, and $\phi:[0,\infty)\rightarrow[0,\infty)$ is a convex function with $\phi(0)=0$, then \[ \int_C\phi(f(x))dx \leq \max_{R}\int_R\phi(g(x))dx. \] The maximum above ranges over the set of truncated cones $R$ with the same volume than $C$ and $g$ is an affine function which becomes zero at one of the bases of $R$ with $g(x_{R,g})=f(x_{C,f})$. Moreover, given such $C$ and $f$, we introduce a new point $x_{C,f}\in C$, which divides evenly the mass distribution of $C$. As a consequence, we obtain a generalization of the type of results [Thm. 1.2, GM] and [Lem. 1.1, MP]. Besides, we also obtain some new estimates on the volume of particular sections of a convex set passing through $x_{C,f}$.