KANTOROVICH POTENTIALS AND CONTINUITY OF TOTAL COST FOR RELATIVISTIC COST FUNCTIONS

KANTOROVICH POTENTIALS AND CONTINUITY OF TOTAL COST FOR RELATIVISTIC COST FUNCTIONS

In this paper we consider the optimal mass transport problem for relativistic cost functions, introduced in [12] as a generalization of the relativistic heat cost. A typical example of such a cost function is ct(x, y) = h(y−x t), h being a strictly convex function when the variable lies on a given b...

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Bibliographic Details
Published in:Journal de mathématiques pures et appliquées
Main Authors: Bertrand, Jérôme, Pratelli, Aldo, Puel, Marjolaine
Format: Journal Article
Language:English
Published: Elsevier 2017
Subjects:
Analysis of PDEs
Mathematics
Metric Geometry
Online Access:Get full text
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Summary:In this paper we consider the optimal mass transport problem for relativistic cost functions, introduced in [12] as a generalization of the relativistic heat cost. A typical example of such a cost function is ct(x, y) = h(y−x t), h being a strictly convex function when the variable lies on a given ball, and infinite otherwise. It has been already proved that, for every t larger than some critical time T > 0, existence and uniqueness of optimal maps hold; nonetheless, the existence of a Kantorovich potential is known only under quite restrictive assumptions. Moreover, the total cost corresponding to time t has been only proved to be a decreasing right-continuous function of t. In this paper, we extend the existence of Kantorovich potentials to a much broader setting, and we show that the total cost is a continuous function. To obtain both results the two main crucial steps are a refined "chain lemma" and the result that, for t > T , the points moving at maximal distance are negligible for the optimal plan.
ISSN:0021-7824