An exterior optimal transport problem

This paper deals with a variant of the optimal transportation problem. Given f $\in$ L 1 (R d , [0, 1]) and a cost function c $\in$ C(R d x R d) of the form c(x, y) = k(y – x), we minimise $\int$ c d$\gamma$ among transport plans $\gamma$ whose first marginal is f and whose second marginal is not prescribed but constrained to be smaller than 1 – f. Denoting by $\mathrm{{\rm Y}}$(f) the infimum of this problem, we then consider the maximisation problem sup{$\mathrm{{\rm Y}}$(f) : $\int$ 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.