On sublinear quasi-metrics and neighborhoods in locally convex cones

Abstract

We consider the topological structure of the sublinear
quasi-metrics in locally convex cones and define the notion of a
locally convex quasi-metric cone. The presence of upper bounded
neighborhoods, gives necessary and sufficient conditions for the
quasi-metrizability of locally convex cones. In particular, we
investigate the boundedness and separatedness of locally convex
quasi-metric cones and characterize the metrizability of locally
convex cones.