标题： 具有无限凸组合的凸性类型函数不等式 显示英文标题

标题： A convexity-type functional inequality with infinite convex combinations

摘要： 给定定义在一个非空且凸的欧几里得空间$d$维子集上的函数$f$，我们证明如果$f$从下方有界并且它满足具有无限凸组合的凸性类型函数不等式，那么$f$必须是凸的。 我们还使用Jensen不等式的概率版本，给出了Daróczy和Páles（1987）以及Pavić（2019）关于无限凸组合下凸性的一些已知结果的推广的替代证明。 显示英文摘要

摘要： 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.