Preprint
Inserted: 4 sep 2023
Last Updated: 4 sep 2023
Year: 2023
Abstract:
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~$m$.
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