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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nonregular extreme points in the set of Minkowski additive selections


Author: Krzysztof Przesławski
Journal: Proc. Amer. Math. Soc. 118 (1993), 1225-1226
MSC: Primary 52A20; Secondary 26B25
MathSciNet review: 1150653
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Abstract: A function $ s:{\mathcal{K}^n} \to {\mathbb{R}^n}$, defined on the family $ {\mathcal{K}^n}$ of all compact convex and nonempty sets in $ {\mathbb{R}^n}$, is called a Minkowski additive selection, provided $ s(A + B) = s(A) + s(B)$ and $ s(A) \in A$, whenever $ A,\;B \in {\mathcal{K}^n}$. We confirm the conjecture [6] that there exist extremal selections which are not regular ($ s$ is regular if $ s\left( A \right) \in \operatorname{ext} A,\;A \in {\mathcal{K}^n}$).


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1150653-4
PII: S 0002-9939(1993)1150653-4
Keywords: Selections, convex sets, extremal points
Article copyright: © Copyright 1993 American Mathematical Society