Compactly convex sets in linear topological spaces

A convex subset $X$ of a linear topological space is called {em compactly convex} if there is a~continuous compact-valued map $Phicolon Xtoexp(X)$ such that $[x,y]subsetPhi(x)cupPhi(y)$ for all $x,yin X$. 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.