Cardinality of some convex sets and of their sets of extreme points
Volume 123 / 2011
Colloquium Mathematicum 123 (2011), 133-147
MSC: 28A12, 28A33, 46A55, 52A07.
DOI: 10.4064/cm123-1-10
Abstract
We show that the cardinality ${\mathfrak n}$ of a compact convex set $W$ in a topological linear space $X$ satisfies the condition that ${\mathfrak n}^{\aleph_0} = {\mathfrak n}$. We also establish some relations between the cardinality of $W$ and that of $\mathop{\rm extr}\nolimits{W}$ provided $X$ is locally convex. Moreover, we deal with the cardinality of the convex set $E(\mu)$ of all quasi-measure extensions of a quasi-measure $\mu$, defined on an algebra of sets, to a larger algebra of sets, and relate it to the cardinality of $\mathop{\rm extr}\nolimits E(\mu)$.
