On Popoviciu's concept of convexity for functions of $d$ variables

TL;DR

This paper develops an integral representation for Popoviciu's convex functions of multiple variables, enabling the derivation of new functional inequalities without requiring differentiability, thus extending prior results to broader function classes.

Contribution

It introduces a general integral representation for Popoviciu's convex functions of multiple variables, broadening the scope beyond differentiable functions and previous two-variable results.

Findings

Derived new functional inequalities for Popoviciu's convex functions.

Extended existing results to non-differentiable functions of multiple variables.

Provided a foundational integral representation for further research.

Abstract

We establish an integral representation for Popoviciu's convex functions of $d$ variables. This representation serves as a~foundation for deriving several functional inequalities, analogous to those well-known for usual convex functions. Our results generalize and extend the results obtained by S.~Gal, C.~Niculescu, B.~Gavrea, T.~Popoviciu, and others, who considered only differentiable functions of two variables. In contrast to other authors, we do not impose any additional regularity assumptions on the studied functions.