Non-archimedean transportation problems and Kantorovich ultra-norms

TL;DR

This paper introduces non-archimedean transportation problems and Kantorovich ultra-norms, extending classical concepts to ultra-metric spaces and non-archimedean fields, with applications to free locally convex spaces.

Contribution

It develops non-archimedean analogs of transportation problems, introduces Kantorovich ultra-norms, and explores free non-archimedean locally convex spaces and their properties.

Findings

Infimum of the NA transportation cost formula is achieved.

Conditions for normability of free NA locally convex spaces are provided.

NA versions of classical theorems like Tkachenko-Uspenskij are established.

Abstract

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.