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Proceedings of the American Mathematical Society

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

 

 

Extremal Minkowski additive selections of compact convex sets


Author: Rade T. Živaljević
Journal: Proc. Amer. Math. Soc. 105 (1989), 697-700
MSC: Primary 52A20
DOI: https://doi.org/10.1090/S0002-9939-1989-0937855-3
MathSciNet review: 937855
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Abstract: A function $ f:{\mathcal{K}^n} \to {R^n}$, defined on the set of all compact convex sets in $ {R^n}$, is a Minkowski additive selection, provided $ f(K + L) = f(K) + f(L)$ and $ f(K) \in K$ for all $ K,L \in {\mathcal{K}^n}$. The paper deals with selections which are extremal in some sense, in particular we characterize the set of all Minkowski additive selections which have the property $ f(K) \in {\text{ext}}(K)$ for all $ K \in {\mathcal{K}^n}$, where $ {\text{ext}}(K)$ is the set of all extreme points of $ K$.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0937855-3
Article copyright: © Copyright 1989 American Mathematical Society