Arc length associated with generalized distance functions

We propose a generalization of the traditional definition of arc length in a manifold. In our definition, the arc length is associated with a distance function d that satisfies the identity property but not necessarily the triangle inequality, non-negativity, definiteness and symmetry. A new class of directed arcs, which we call “d-conservative” arcs, arises in an evident manner from our definition. These arcs satisfy a property of conservation of the d-distance along the arc. Each d-conservative arc has a d-length equal to the d-distance between its endpoints. If d satisfies the triangle inequality, the d-conservative arcs coincide with the arcs of minimum d-length. We prove that the d-length of an arc can be expressed as the integral of a function F along the arc, where F is the one-sided directional derivative of d. This last relation between d and F was proved by Busemann and Mayer (1941) [3] for the Finsler distances, which satisfy, among others, the triangle inequality and non-negativity, requirements that we do not need in our proof. We also prove that if the one-sided directional derivative F of a distance function d is continuous, then d satisfies the triangle inequality if, and only if, F is convex. We analyze an example of a non-positive definite distance function.