Weight of convex compact spaces and their boundaries

Abstract

It is proved that if some boundary $B$ of a convex compact subset $X$ of a locally convex linear space has a countable network, then the convex compact space $X$ is metrizable. If the boundary $B$ is a Lindelof $\Sigma$-space, then the network weight $nw(B)$ of $B$ coincides with the weight $w(K)$ of $K$.