On Popoviciu-Ionescu functional equation

TL;DR

This paper presents a novel solution to the Popoviciu-Ionescu functional equation using a higher-dimensional distribution generalization, extending previous solutions and proposing new open problems.

Contribution

It introduces a new approach based on distribution theory in higher dimensions, differing from prior solutions to the functional equation.

Findings

New solution method for the functional equation

Extension of Radó's theorem to higher dimensions

Proposes open problems for future research

Abstract

We study a functional equation first proposed by T. Popoviciu in 1955. It was solved for the easiest case by Ionescu in 1956 and, for the general case, by Ghiorcoiasiu and Roscau, and Rad\'o in 1962. Our solution is based on a generalization of Rad\'o's theorem to distributions in a higher dimensional setting and, as far as we know, is different than existing solutions. Finally, we propose several related open problems.