An exterior optimal transport problem

This paper deals with a variant of the optimal transportation problem. Given f∈L1(Rd,[0,1]) and a cost function c∈C(Rd×Rd) of the form c(x,y)=k(y-x), we minimise ∫cdγ among transport plans γ whose first marginal is f and whose second marginal is not prescribed but constrained to be smaller than 1-f. Denoting by Υ(f) the infimum of this problem, we then consider the maximisation problem sup{Υ(f):∫f=m} where m>0 is given. We prove that maximisers exist under general assumptions on k, and that for k radial, increasing and coercive these maximisers are the characteristic functions of the balls of volume m.