A convexity-type functional inequality with infinite convex combinations

TL;DR

This paper proves that functions satisfying a specific convexity-type inequality with infinite convex combinations are necessarily convex, providing new proofs for related generalized convexity results using probabilistic Jensen inequality.

Contribution

It establishes that bounded functions meeting a convexity-type inequality with infinite combinations are convex and offers alternative proofs for existing generalized convexity theorems.

Findings

Functions with the inequality are convex

Provides probabilistic proof techniques

Generalizes previous convexity results

Abstract

Given a function $f$ defined on a nonempty and convex subset of the $d$-dimensional Euclidean space, we prove that if $f$ is bounded from below and it satisfies a convexity-type functional inequality with infinite convex combinations, then $f$ has to be convex. We also give alternative proofs of a generalization of some known results on convexity with infinite convex combinations due to Dar\'oczy and P\'ales (1987) and Pavi\'c (2019) using a probabilistic version of Jensen inequality.