This paper deals with a variant of the optimal transportation problem. Given \(f \in L^1( \mathbb {R}^d, [0,1])\) and a cost function \(c \in C(\mathbb {R}^d \times \mathbb {R}^d)\) of the form \(c(x,y)=k(y-x)\) , we minimise \( \smallint c \,d\gamma \) 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 \(\Upsilon (f)\) the infimum of this problem, we then consider the maximisation problem \(\sup \{\Upsilon (f): \, \smallint f = m \}\) 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.
