Jensen-type geometric shapes

TL;DR

This paper characterizes convex shapes for which the average value of any convex function over the shape is less than or equal to the average over its boundary, providing necessary and sufficient conditions.

Contribution

It establishes new geometric criteria for Jensen-type inequalities to hold for convex shapes, including parallelotopes, balls, and certain polytopes.

Findings

Inequality holds for parallelotopes, balls, and specific polytopes with inscribed spheres.

Provides necessary and sufficient conditions for the inequality to be valid.

Characterizes shapes where boundary averages dominate volume averages for convex functions.

Abstract

We present both necessary and sufficient conditions to the convex closed shape $X$ such that the inequality $$ \frac{1}{|X|} \int_X f(x)\:dx \le \frac{1}{|\partial X|} \int_{\partial X} f(x)\:dx$$ is valid for every convex function $f \colon X \to \mathbb{R}$ ($\partial X$ stands for the boundary of $X$). It is proved that this inequality holds if $X$ is (i) an $n$-dimensional parallelotope, (ii) an $n$-dimensional ball, (iii) a convex polytope having an inscribed sphere (tangent to all its facets) with center in the center of mass of $\partial X$.