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arXiv · 2508.02474

A convexity-type functional inequality with infinite convex combinations

Matyas Barczy·Zsolt Páles
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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.

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Matyas Barczy, Zsolt Páles. 2025-08-04. A convexity-type functional inequality with infinite convex combinations. https://arxiv.org/abs/2508.02474

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