Evgenii Aleksandrovich Reznichenko

Weight of convex compact sets and their boundaries.

The boundary of a convex compact set $K$ is a set $B\subset K$ such that any continuous affine function on $K$ attains a maximum on $B$ . The set of extreme points is the boundary. It has been proved that if some boundary $B$ of a convex compact subset $K$ of a locally convex linear space has a countable network, then the convex compact set $K$ is metrizable. R. Haydon proved this statement in 1976 for the case when $B$ is a set of extreme points. J. Spurnỳ proved this statement in 2010 for the case when $B$ is a separable metrizable space. If the boundary $B$ is a Lindelof $\Sigma$ -space, then the network weight $nw(B)$ of the boundary $K$ coincides with the weight $w(K)$ of the convex compact set $K$ .

Algebra, Topology, and Dynamical Systems

A conference in honor of Dikran Dikranjan on the occasion of his 75th birthday and of Ivan Prodanov on the occasion of the 90th anniversary of his birth

Evgenii Aleksandrovich Reznichenko