On Popoviciu’s Concept of Convexity for Functions of d Variables

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.