 
Summary: SCUOLA NORMALE SUPERIORE
Pisa
Preprints di Matematica  n. 9 Aprile 2004
Continuity of an optimal
transport in monge problem
I. FRAGALĄ  M. S. GELLI  A. PRATELLI
Stampato in proprio
APRILE 2004
Scuola Normale Superiore, Piazza dei Cavalieri 7, PISA
SNSMathPreprintServerhttp://math.sns.it/papers/fragelpra04/
CONTINUITY OF AN OPTIMAL TRANSPORT IN MONGE PROBLEM
ILARIA FRAGAL`A, MARIA STELLA GELLI, AND ALDO PRATELLI
Abstract. Given two absolutely continuous probability measures f± in R2, we consider the classical
Monge transport problem, with the Euclidean distance as cost function. We prove the existence of a
continuous optimal transport, under the assumptions that (the densities of) f± are continuous and
strictly positive in the interior part of their supports, and that such supports are convex, compact, and
disjoint. We show through several examples that our statement is nearly optimal. Moreover, under the
same hypotheses, we also obtain the continuity of the transport density.
Keywords. Monge problem, optimal transport, continuity, maximal transport rays.
MSC2000. 49N60, 28A50, 90B06, 74P20.
