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)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}$).

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

  • [1] Z. Artstein, Stabilizing selections of differential inclusions, Weizmann Institute of Science, preprint.
  • [2] Cz. Olech, A note concerning extremal points of a convex set, Bull. Acad. Sci. Ser. Sci. Math. Fis. Nat. 13 (1965), 347-352. MR 0187144 (32:4598)
  • [3] K. Przesławski, Faces of convex sets and Minkowski additive selections, unpublished.
  • [4] K. Przesławski and D. Yost, Continuity properties of selectors and Michael's Theorem, Michigan Math. J. 36 (1989), 113-134. MR 989940 (90d:49010)
  • [5] G. Stefani and P. Zecca, Multivalued differential equations on manifolds with application to control theory, Illinois Math. J. 24 (1980), 560-575. MR 586796 (84b:58101)
  • [6] R. Živaljević, Extremal Minkowski additive selections of compact convex sets, Proc. Amer. Math. Soc. 105 (1989), 697-700. MR 937855 (89g:52002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52A20, 26B25

Retrieve articles in all journals with MSC: 52A20, 26B25

Additional Information

Keywords: Selections, convex sets, extremal points
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society