Compactly convex sets in linear topological spaces

Author T. Banakh, M. Mitrofanov, O. Ravsky
tbanakh@yahoo.com
Ivan Franko National University of Lviv (Ukraine) and Jan Kochanowski University, Kielce (Poland); Institute of Applied Problems of Mathematics and Mechanics of Ukrainian Academy of Science, Lviv, Ukraine

Abstract A convex subset $X$ of a linear topological space is called {\em compactly convex} if there is a~continuous compact-valued map $\Phi\colon X\to\exp(X)$ such that $[x,y]\subset\Phi(x)\cup\Phi(y)$ for all $x,y\in X$. We prove that each convex subset of the plane is compactly convex. On the other hand, the space $\mathbb{R}^3$ contains a convex set that is not compactly convex. Each compactly convex subset $X$ of a linear topological space $L$ has locally compact closure $\bar X$ which is metrizable if and only if each compact subset of $X$ is metrizable.
Keywords compactly convex set; linear topological space
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Pages 161-173
Volume 37
Issue 2
Year 2012
Journal Matematychni Studii
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