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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On points at which a set is cone-shaped
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by M. Edelstein, L. Keener and R. O’Brien PDF
Proc. Amer. Math. Soc. 66 (1977), 327-330 Request permission

Abstract:

A set $\mathcal {S}$ in a normed linear space X is said to be cone-shaped at $x \in X$ if there is a closed half-space that has x in its bounding hyperplane and contains $\{ y \in \mathcal {S}:[x,y] \subset S\}$. The point x is called a cone point. In this paper it is shown that if X has an equivalent uniformly convex and uniformly smooth norm and if $\mathcal {S}$ is a closed bounded subset with the finite visibility property for cone points (i.e., for every finite set F of cone points of S there is a point $z \in S$ such that $[z,y] \subset \mathcal {S}$ for all $y \in F$), then S is starshaped.
References
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 66 (1977), 327-330
  • MSC: Primary 46B05; Secondary 52A05
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0454593-6
  • MathSciNet review: 0454593