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Proceedings of the American Mathematical Society

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

 

 

Some remarks of drop property


Author: Pei-Kee Lin
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
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1095224-2
Keywords: Drop property, reflexive space, compact, nearly uniformly convex
Article copyright: © Copyright 1992 American Mathematical Society