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arXiv · 2309.02806

An exterior optimal transport problem

Jules Candau-Tilh·Michael Goldman·Benoît Merlet
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Abstract

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.

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Jules Candau-Tilh, Michael Goldman, Benoît Merlet. 2023-09-06. An exterior optimal transport problem. https://arxiv.org/abs/2309.02806

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