Popoviciu's inequality for functions of several variables

T. Popoviciu (1965) [13] has proved an interesting characterization of the convex functions of one real variable, based on an inequality relating the values at any three points x1,x2,x3, with the values at their means of different 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.