Compactly convex sets in linear topological spaces

A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\Phi:X\to exp(X)$ 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 $R^3$ 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.

💡 Research Summary

The paper introduces the notion of a “compactly convex” subset of a linear topological space. A convex set (X\subset L) is called compactly convex if there exists a continuous compact‑valued map (\Phi:X\to\exp(X)) (where (\exp(X)) denotes the hyperspace of non‑empty compact subsets of (X) equipped with the Vietoris topology) such that for every pair of points (x,y\in X) the closed line segment (