Optimal measure transportation with respect to non-traditional costs

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