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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.

Balayage defined by the nonnegative convex functions