Monge properties, discrete convexity and applications

Starting from Monge's mass transportation problem we review the role Monge properties play in optimization. In particular we discuss transportation problems whose cost functions fulfill a Monge property, Monge sequences, algebraic Monge properties, the recognition of permuted Monge arrays and multidimensional Monge arrays and the connections between Monge properties and discrete convexity. Finally we discuss Prékopa's recent approach using Monge arrays in bounding multivariate probability distribution functions.