  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 $\Upsilon$(f) the
infimum of this problem, we then consider the maximisation problem
sup{$\Upsilon$(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.
