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.