Kantorovich's Mass Transport Problem for Capacities

TL;DR

This paper extends Kantorovich's mass transport problem to capacities, establishing duality and cyclic monotonicity of optimal plans within this generalized framework.

Contribution

It introduces a novel extension of mass transport theory to capacities and proves key properties like duality and cyclic monotonicity in this setting.

Findings

Extension of Kantorovich's problem to capacities.

Proof of cyclic monotonicity of optimal plans.

Establishment of duality in the capacity framework.

Abstract

The aim of the present paper is to extend Kantorovich's mass transport problem to the framework of upper/lower continuous capacities and to prove the cyclic monotonicity of the supports of optimal supermodular plans. As in the probabilistic case, this easily yields the corresponding extension of the Kantorovich duality.