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Non-archimedean transportation problems and Kantorovich ultra-norms

Authors
  • Megrelishvili, Michael
  • Shlossberg, Menachem
Type
Preprint
Publication Date
Apr 12, 2016
Submission Date
Apr 23, 2015
Identifiers
arXiv ID: 1504.06301
Source
arXiv
License
Yellow
External links

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.

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