Attainable separation property and asymptotic hyperplane for a closed convex set in a normed space

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:489

Article

Zheng, Xi Yin¹ 

[1] Yunnan Univ, Dept Math, Kunming 650091, Yunnan, Peoples R China

关键词: Separation theorem;    Asymptotic hyperplane;    Reflexive Banach space;   

DOI  :  10.1016/j.jmaa.2020.124121

来源: Elsevier

【 摘 要 】

Given a closed convex set A in a normed space X, motivated by the classical separation theorem and Gale and Klee's strict separation property, this paper introduces attainable separation property and attainable strict separation property of A. It is proved that if A has attainable separation property then S(A, x*) = {a is an element of A vertical bar < x*, a > = supx < x*, x >}(x is an element of A) is nonempty and bounded for all x* is an element of X*\{0} with supx < x*, x >(x is an element of A) < infinity, which reduces to the James theorem if A is bounded. A geometrical notion of an asymptotic hyperplane for A is adopted to study attainable separation properties. It is proved that A has no asymptotic hyperplane if and only if it is continuous in Gale and Klee's sense and that, under the reflexivity assumption on X, A has no asymptotic hyperplane if and only if A - B is closed for every closed convex set Bin X with int (B) not equal phi. Moreover, it is proved that if Xis not reflexive then such difference A - B is not necessarily closed. Using the techniques of variational analysis and in terms of asymptotic hyperplanes, several characterizations and sufficient conditions are established for A to have attainable separation property and attainable strict separation property. In the case of finite dimensional spaces, some sharper results are established. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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