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.