Jensen-type geometric shapes

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 $f:X\to \mathbb{R}$ ($\partial X$ 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 $\partial X$.

Cited by in corpus (1)

• The Hermite-Hadamard inequality in higher dimensions