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Strong Minkowski separation and co-drop property. (English) Zbl 1144.46007

The present article is inspired by [L.X.Cheng, Y.C.Zhou, and F.Zhang, Proc.Am.Math.Soc.124, No.12, 3699–3702 (1996; Zbl 0863.46003)]. A subset \(A\) of a locally convex space \(X\) has the co-drop property provided that for each closed convex subset \(B\) of \(X\) with \(0\notin \text{cl}(A-B)\) there is \(a\in A\) satisfying \(D(a,B)\cap A=\{a\}\). As usual, \(D(a,B)=\text{co}(\{a\}\cup B)\) is the drop from \(a\) to \(B\). The author shows that a bounded convex set \(A\) has the co-drop property if and only if each continuous linear functional on \(X\) attains its maximum on \(A\). Some relevant matters are discussed.

MSC:

46A55 Convex sets in topological linear spaces; Choquet theory
46A03 General theory of locally convex spaces

Citations:

Zbl 0863.46003
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References:

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