The Monge Problem in Banach Spaces

In this paper, we generalize the Kantorovich functional to Köthe-spaces for a cost or a profit function. We examine the convergence of probabilities with respect to this functional for some Köthe-spaces. We study the Monge problem: Let $$\mathbb{E}$$ be a Köthe-space, P and Q two Borel probabilities defined on a Polish space M and a cost function $$c: M \times M \to \mathbb{R}_{+}$$ . A Köthe functional $$\mathcal{I}$$ is defined by $$\mathcal{I}$$ (P, Q) = inf $$\{\|c(X, Y)\|; \mathcal{L}(X) = P, \mathcal{L}(Y) = Q\}$$ where $$\mathcal{L}(X)$$ is the law of X. If c is a profit function, we note $$\mathcal{S}$$ . (P, Q) = sup $$\{\|c(X,Y)\|,\mathcal{L}(X) = P, \mathcal{L}(Y) = Q\}$$ Under some conditions, we show the existence of a Monge function, φ, such that $$\mathcal{I}(P, Q) = \|c(X, \phi (X))\|$$ , or $$\mathcal{S}(P,Q)=\|c(X, \phi(X))\|, \mathcal{L}(X)=P, \mathcal{L}(\phi(X))=Q$$ .