Non-archimedean transportation problems and Kantorovich ultra-norms

We study a non-archimedean (NA) version of transportation problems and introduce naturally arising ultra-norms which we call Kantorovich ultra-norms. For every ultra-metric space and every NA valued field (e.g., the field $\mathbb Q_{p}$ of $p$-adic numbers) the naturally defined inf-max cost formula achieves its infimum. We also present NA versions of the Arens-Eells construction and of the integer value property. We introduce and study free NA locally convex spaces. In particular, we provide conditions under which these spaces are normable by Kantorovich ultra-norms and also conditions which yield NA versions of Tkachenko-Uspenskij theorem about free abelian topological groups.