Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces

Abstract

In this paper we extend the duality theory of the multi-marginal optimal transport problem for cost functions depending on a decreasing function of the distance (not necessarily bounded). This class of cost functions appears in the context of SCE Density Functional Theory introduced in Strong-interaction limit of density-functional theory by M. Seidl [23].

Multi-marginal entropy-transport with repulsive cost

“…We notice that the set Π sym N (ρ) is compact in the w * -topology [18]. The functional E is lower semicontinuous by Corollary 2.4, and in our setting the lower semicontinuity of C 0 is proven as a part of the proof of [15,Proposition 3.1]. Since for each ε ≥ 0 the functional C ε is convex, we conclude that it has a minimizer in the set Π sym N (ρ) 2.2.…”

“…The next theorem from [4] (see also [15,Theorem 3.2]) states that for measures ρ satisfying the assumptions (A) and (B) there exists α > 0 for which the support of any optimal plan is concentrated away from the set…”

Continuity of Multimarginal Optimal Transport with Repulsive Cost

“…the proof is carried out for the Coulomb cost but adapts to the assumption that c is lower continuous; moreover, the existence of an optimal potential \varphi in the dual formulation (1.5) is also proved (see also [17] for a generalization to costs not necessarily bounded from below). We remark that a general duality result has already been proven by Kellerer in [22], but the hypothesis on the cost function could not be adapted to a Coulomb-type cost.…”