Popoviciu's inequality for functions of several variables

Abstract. T. Popoviciu [13] has proved in 1965 an interesting characteriza-tion of the convex functions of one real variable, based on an inequality relat-ing the values at any three points x1; x2; x3; with the values at their means of di¤erent orders: (x1 + x2) =2; (x2 + x3) =2; (x3 + x1) =2 and (x1 + x2 + x3) =3. The aim of our paper is to develop a higher dimensional analogue of the usual convexity based on his characterization. 1