Closedness of bounded convex sets of asymmetric normed linear spaces and the Hausdorff quasi-metric

Abstract

If $A$ is a (nonempty) bounded convex subset of an asymmetric normed linear space $(X,q),$ we define the closedness of $A$ as the set \textrm{cl}$% _{q}A\cap \mathrm{cl}_{q^{-1}}A,$ and denote by $CB_{0}(X)$ the collection of the closednesses of all (nonempty) bounded convex subsets of $(X,q).$ We show that $CB_{0}(X),$ endowed with the Hausdorff quasi-metric of $q,$ can be structured as a quasi-metric cone. Then, and extending a classical embedding theorem of L. H\"{o}rmander, we prove that there is an isometric isomorphism from this quasi-metric cone into the product of two asymmetric normed linear spaces of bounded continuous real functions equipped with the asymmetric norm of uniform convergence.