Some remarks of drop property
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- by Pei-Kee Lin PDF
- Proc. Amer. Math. Soc. 115 (1992), 441-446 Request permission
Abstract:
Let $C$ be a proper closed convex set. $C$ is said to have the drop property if for any nonempty closed set $A$ disjoint with $C$, there is $a \in A$ such that ${\text {co(}}a,C{\text {)}} \cap {\text {A = }}\left \{ a \right \}$. We show that if $X$ contains a noncompact set with the drop property, then $X$ is reflexive. Moreover, we prove that if $C$ is a noncompact closed convex subset of a reflexive Banach space, then $C$ has the drop property if and only if $C$ satisfies the following conditions: (i) the interior of $C$ is nonempty; (ii) $C$ does not have any asymptote, and the boundary of $C$ does not contain any ray; and (iii) every support point $x$ of $C$ is a point of continuity.References
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Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 441-446
- MSC: Primary 46B20; Secondary 52A07
- DOI: https://doi.org/10.1090/S0002-9939-1992-1095224-2
- MathSciNet review: 1095224