On Popoviciu-Ionescu functional equation

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.