KANTOROVICH POTENTIALS AND CONTINUITY OF TOTAL COST FOR RELATIVISTIC COST FUNCTIONS

Jérôme Bertrand (1) , Aldo Pratelli (2) , Marjolaine Puel (1, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Department Mathematik [Erlangen]
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

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.

Domaines: Géométrie métrique [math.MG]; Equations aux dérivées partielles [math.AP]

Jérôme Bertrand, Aldo Pratelli, Marjolaine Puel. KANTOROVICH POTENTIALS AND CONTINUITY OF TOTAL COST FOR RELATIVISTIC COST FUNCTIONS. Journal de Mathématiques Pures et Appliquées, 2017. ⟨hal-01967247⟩