Marginals with Finite Repulsive Cost

Abstract

We consider a multimarginal transport problem with repulsive cost, where the marginals are all equal to a fixed probability ρ ∈ P(R d ). We prove that, if the concentration of ρ is less than 1/N , then the problem has a solution of finite cost. The result is sharp, in the sense that there exists ρ with concentration 1/N for which the cost is infinite.

On c-cyclical monotonicity for optimal transport problem with Coulomb cost

“…For transport costs which tend to +∞ when |x i − x j | → 0 the concentration of the datum ρ plays an important role in the finiteness and the continuity of the cost and in properties of optimal transport plans [2,4,12,38]. For ρ ∈ P(R d ) we introduce a measure of the concentration of ρ at the scale r µ ρ (r) = sup…”

Continuity of Multimarginal Optimal Transport with Repulsive Cost

“…We acknowledge here also the recent paper [2] in which the first two parts of the theorem above are proved by a slicing method and by induction on the number of atoms. Our proof is considerably shorter, and it is based instead on a dimension reduction argument, which allows us to reduce the problem to a 1-dimensional problem via a suitably chosen projection.…”

“…The article is a refinement of the results presented in [5], especially from the point of view of the assumptions, which in our work are shown to be sharp. We acknowledge also the recent preprint [2] in which the finiteness of the cost is proved in a similar fashion by dimension reduction.…”

“…Besides exploring the connection with density functional theory [6,10,3,20], several authors investigated the mathematical properties of the minimizer. A natural question is whether the optimizer is induced by a map, namely if there exists a Borel map T : X \rightar X such that T \# \rho = \rho (where T \# \rho represents the pushforward measure of the measure \rho through the Borel map T ) and an optimizer \pi in the minimization problem in (1.4) can be represented as (Id, T, T (2) , . .…”