Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Balayage defined by the nonnegative convex functions

Authors: P. Fischer and J. A. R. Holbrook
Journal: Proc. Amer. Math. Soc. 79 (1980), 445-448
MSC: Primary 46A55; Secondary 26B25
MathSciNet review: 567989
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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}, \ldots ,{x_r},{y_1}, \ldots ,{y_n}$, where $ r \leqslant n$, be points in $ {{\mathbf{R}}^d}$. Then

$\displaystyle \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

$\displaystyle {x_j} = \sum\limits_{k = 1}^n {{m_{jk}}{y_k}\quad (j = 1, \ldots ,r).} $

In the case $ 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.

References [Enhancements On Off] (What's this?)

  • [1] D. Blackwell, Equivalent comparisons of experiments, Ann. Math. Statistics 24 (1953), 265-272. MR 0056251 (15:47b)
  • [2] P. Fischer and J. A. R. Holbrook, Matrices sous-stochastiques et fonctions convexes, Canad. J. Math. 29 (1977), 631-637. MR 0625521 (58:30042)
  • [3] P. A. Meyer, Probability and potentials, Blaisdell, Waltham, Mass., 1966. MR 0205288 (34:5119)
  • [4] L. Mirsky, Majorization of vectors and inequalities for convex functions, Monatsh. Math. 65 (1961), 159-169. MR 0123661 (23:A986)
  • [5] R. R. Phelps, Lectures on Choquet's theorem, Van Nostrand, Princeton, N. J., 1966. MR 0193470 (33:1690)
  • [6] W. Rudin, Functional analysis, McGraw-Hill, New York, 1973. MR 0365062 (51:1315)
  • [7] S. Sherman, On a theorem of Hardy, Littlewood, Pólya and Blackwell, Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 826-831. MR 0045787 (13:633g)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46A55, 26B25

Retrieve articles in all journals with MSC: 46A55, 26B25

Additional Information

Keywords: Hardy-Littlewood-Pólya order, doubly stochastic matrices, balayage
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society