Abstract

SANCHEZ-LARIOS, H.  and  GUILLEN-BURGUETE, S.T.. Asymmetric and non-positive definite distance functions Part I: Theoretical framework. Ing. invest. y tecnol. [online]. 2008, vol.9, n.4, pp.339-346. ISSN 2594-0732.

We propose a theoretical framework for modeling generalized distance functions, which can be asymmetric and non-positive definite. We give a definition of arc length associated to a generalized distance function d. Our distance function d satisfies the identity property but, unlike metrics, may not satisfy the triangle inequality, or symmetry and definiteness properties. We show that each distance function d induces certain arcs, called "d-induced", which satisfy a conservation law of the distance d and are a generalization of the straight line segments of the Euclidean space. We also show that if d satisfies the triangle inequality, then the d-induced arcs are arcs of minimal length with respect to the distance function d, and in this case, the distance function d can be modeled as a problem of calculus of variations.

Keywords : Generalized distance functions; length; triangle inequality; Finsler metric.