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Proceedings of the American Mathematical Society
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
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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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0567989-9
PII: S 0002-9939(1980)0567989-9
Keywords: Hardy-Littlewood-Pólya order, doubly stochastic matrices, balayage
Article copyright: © Copyright 1980 American Mathematical Society