The convex hull and the Caratheodory number of a set in terms of the metric projection operator

We prove that each point of the convex hull of a compact set $M$ in a smooth Banach space $X$ can be approximated arbitrarily well by convex combinations of best approximants from $M$ to $x$ (values of the metric projection operator $P_M(x)$), where $x \in X$. As a corollary, we show that the Caratheodory number of a compact set $M \subset X$ with at most $k$-valued metric projection $P_M$ is majorized by $k$, that is, each point in the convex hull of $M$ lies in the convex hull of at most $k$ points of $M$.Bibliography: 26 titles.

metric projection, convex hull, Banach space, smoothness, Minkowski functional, Caratheodory number