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 $Φ:X\to exp(X)$ such that $[x,y]\subsetΦ(x)\cup Φ(y)$ 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.

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