Popoviciuʹs 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 x 1 , x 2 , x 3 , with the values at their means of different orders: ( x 1 + x 2 ) / 2 , ( x 2 + x 3 ) / 2 , ( x 3 + x 1 ) / 2 and ( x 1 + x 2 + x 3 ) / 3 . The aim of our paper is to develop a higher dimensional analogue of the usual convexity based on his characterization.