Non-archimedean transportation problems and Kantorovich ultra-norms

M. Megrelishvili; M. Shlossberg

doi:https://doi.org/10.1134/S2070046616020035

Non-archimedean transportation problems and Kantorovich ultra-norms

M. Megrelishvili, M. Shlossberg

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

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 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.

Original language	English
Pages (from-to)	125-148
Number of pages	24
Journal	P-Adic Numbers, Ultrametric Analysis, and Applications
Volume	8
Issue number	2
DOIs	https://doi.org/10.1134/S2070046616020035
State	Published - 1 Apr 2016

Keywords

Fermat-Torricelli point
Kantorovich ultra-norm
Kantorovich-Rubinstein seminorm
Levi-Civita field
free abelian topological group
free locally convex space
non-archimedean valuation
p-adic numbers
transportation problem
valued field

All Science Journal Classification (ASJC) codes

General Mathematics

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Cite this

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title = "Non-archimedean transportation problems and Kantorovich ultra-norms",

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 Qp 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.",

keywords = "Fermat-Torricelli point, Kantorovich ultra-norm, Kantorovich-Rubinstein seminorm, Levi-Civita field, free abelian topological group, free locally convex space, non-archimedean valuation, p-adic numbers, transportation problem, valued field",

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AU - Megrelishvili, M.

AU - Shlossberg, M.

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N2 - 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 Qp 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.

AB - 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 Qp 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.

KW - Fermat-Torricelli point

KW - Kantorovich ultra-norm

KW - Kantorovich-Rubinstein seminorm

KW - Levi-Civita field

KW - free abelian topological group

KW - free locally convex space

KW - non-archimedean valuation

KW - p-adic numbers

KW - transportation problem

KW - valued field

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M3 - مقالة

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SP - 125

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JO - P-Adic Numbers, Ultrametric Analysis, and Applications

JF - P-Adic Numbers, Ultrametric Analysis, and Applications

IS - 2

ER -

Non-archimedean transportation problems and Kantorovich ultra-norms

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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