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Tietze-type theorem for locally nonconical convex sets. (English) Zbl 0964.46004

Author’s abstract: A convex subset \(Q\) of a real topological linear space \(L\) is called locally nonconical at a point \(q\in Q\) if and only if there exists a relative neighbourhood \(Q_q\) of \(q\) in \(Q\) such that for every two points \(x,y\in Q_q\) there is a relative neighbourhood \(Q_x\) of \(x\) in \(Q\) such that \(Q_x+{1\over 2}(y- x)\subseteq Q\). \(Q\) is called locally nonconical if and only if this condition is satisfied for every two points \(x,y\in Q\). It is proved that \(Q\) is locally nonconical if and only if it is locally nonconical at every boundary point belonging to \(Q\). This contributes to a recent work of Shell.

MSC:

46A55 Convex sets in topological linear spaces; Choquet theory
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
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