Optimal measure transportation with respect to non-traditional costs

Abstract

We study optimal mass transport problems between two measures with respect to a non-traditional cost function, i.e. a cost c which can attain the value \(+\infty \). We define the notion of c-compatibility and strong c-compatibility of two measures, and prove that if there is a finite-cost plan between the measures then the measures must be c-compatible, and if in addition the two measures are strongly c-compatible, then there is an optimal plan concentrated on a c-subgradient of a c-class function. This function is the so-called potential of the plan. We give two proofs of this theorem, under slightly different assumptions. In the first we utilize the notion of c-path-boundedness, showing that strong c-compatibility implies a strong connectivity result for a directed graph associated with an optimal map. Strong connectivity of the graph implies that the c-cyclic monotonicity of the support set (which follows from classical reasoning) guarantees its c-path-boundedness, implying, in turn, the existence of a potential. We also give a constructive proof, in the case when one of the measures is discrete. This approach adopts a new notion of ‘Hall polytopes’, which we introduce and study in depth, to which we apply a version of Brouwer’s fixed point theorem to prove the existence of a potential in this case.