On a Generalization of the Hermite–Hadamard Inequality and Applications in Convex Geometry

Abstract

In this paper, we show the following result: if C is an n-dimensional 0-symmetric convex compact set, $$f:C\rightarrow [0,\infty )$$ is concave, and $$\phi :[0,\infty )\rightarrow [0,\infty )$$ is not identically zero, convex, with $$\phi (0)=0$$ , then $$\begin{aligned} \frac{1}{|C|}\int _C\phi (f(x)){\text {d}}x\le \frac{1}{2}\int _{-1}^1\phi (f(0)(1+t)){\text {d}}t, \end{aligned}$$ where |C| denotes the volume of C. If $$\phi $$ 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.