2021/04/10 by Artstein-Avidan, Shiri, Sadovsky, Shay, Wyczesany, Katarzyna · 1 citation
#49K27 #49Q22 #52A01 #52A23 #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC) #Probability (math.PR)
paper · doi:10.48550/arxiv.2104.04838
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 +∞. 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.