On sublinear quasi-metrics and neighborhoods in locally convex cones

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.