id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-38	Bansil, Mohit; Kitagawa, Jun	An optimal transport problem with storage fees	2023	24	.pdf	application/pdf	11780	573	76	Since F ∗∗ ≤ F , we also have mF∗∗ < ∞. Then by strong duality combined with Proposition 5.1 below, we see (since F ∗ = F ∗∗∗) 0 By [9, Corollary 13.3.3] and since F ∗∗∗ = F ∗, we see that F ∗∗ also has Lipschitz constant L, then a calculation similar to (6.2) with F ∗∗ replacing F shows that the EJDE-2023/22 AN OPTIMAL TRANSPORT PROBLEM 23 pair (T, λ∞) minimizes (1.2) with storage fee function F ∗∗. Thus, by the strong duality Theorem 3.3 we have mF −MF = mF −mF∗∗ = F (λ∞)− F ∗∗(λ∞)	cache/ejde-38.pdf	txt/ejde-38.txt