$L^\infty$-optimal transport for a class of quasiconvex cost functions

TL;DR

This paper investigates the $L^ Infty$-optimal transport problem for a new class of quasiconvex cost functions, establishing conditions for the existence and uniqueness of Monge minimizers and introducing a twist condition.

Contribution

It introduces a new class of cost functions with a tentative twist condition, analyzing minimizer properties and uniqueness in the $L^ Infty$-optimal transport setting.

Findings

Conditions under which minimizers are induced by transportation maps

Uniqueness of cyclically monotone Monge minimizers for the new cost class

Comparison with previous results in optimal transport literature

Abstract

We consider the $L^\infty$-optimal mass transportation problem \[ \min_{\Pi(\mu, \nu)} \gamma-\mathrm{ess\,sup\,} c(x,y), \] for a new class of costs $c(x,y)$ for which we introduce a tentative notion of twist condition. In particular we study the conditions under which the infinitely-motonone minimizers are induced by a transportation map. We also state a uniqueness result for infinitely cyclically monotone Monge minimizers that corresponds to this class of cost functions. We compare the results to previous works.