Balayage defined by the nonnegative convex functions

Abstract

We study the Choquet order induced on measures on a linear space by the cone of nonnegative convex functions. We are concerned mainly with discrete measures, and the following result is typical. Let x 1 , … , x r , y 1 , … , y n {x_1}, \ldots ,{x_r},{y_1}, \ldots ,{y_n} , where r ⩽ n r \leqslant n , be points in R d {{\mathbf {R}}^d} . Then \[ ∑ 1 r f ( x k ) ⩽ ∑ 1 n f ( y k ) \sum \limits _1^r {f({x_k}) \leqslant } \sum \limits _1^n {f({y_k})} \] for all nonnegative, continuous, convex functions f if, and only if, there exists a doubly stochastic matrix M such that \[ x j = ∑ k = 1 n m j k y k ( j = 1 , … , r ) . {x_j} = \sum \limits _{k = 1}^n {{m_{jk}}{y_k}\quad (j = 1, \ldots ,r).} \] In the case d = 1 d = 1 , this result may be found in the work of L. Mirsky; our methods allow us to place such results in a general setting.