A method for modeling generalized distance functions

We propose a method for modeling distance functions in an n dimensional Euclidean space such that, for each ordered pair of distinct points there is at least one path connecting them. Our distance functions may refer to transportation cost, travel distance, travel time, energy expended, etc. We introduce two new concepts: generalized distance function (gdf) and are induced by a gdf. A gdf, unlike metrics, can be asymmetric and nonpositive definite, and unlike Lp metrics, it can be nonuniform. We show that a gdf can be obtained by solving a problem of the calculus of variations, where the value of the functional for a given arc joining two points represents the length of the arc measured with respect to that gdf. We obtain a gdf with physical interpretation.