A Theoretical Framework for Modeling Asymmetric, Nonpositive Definite and Nonuniform Distance Functions on Rⁿ

H. Sánchez–Larios*¹, S. Guillén–Burguete²

^1,2 Instituto de Ingeniería, Universidad Nacional Autónoma de México Edificio 12 del Instituto de Ingeniería, Circuito Exterior, Ciudad Universitaria, 04360 México, D. F *E–mail: herica.sanchez@ciencias.unam.mx

ABSTRACT

In this paper, we give theoretical foundations for modeling distance functions on the usual Euclidean space Rⁿ, where distance may refer to physical kilometers, liters of fuel consumed, time spent in traveling, or transportation cost. In our approach, a distance function d is derived from a function F₀ called the fundamental function of d. Our distance functions, unlike metrics, can be asymmetric and non–positive definite, and unlike the Lp norms, they can be nonuniform. We illustrate our theoretical framework by modeling an asymmetric and non–uniform distance function on R² which can take negative values.

Keywords: Distance function, isotropy, non–uniformity, asymmetry, weighted Lp–norm.

RESUMEN

En este artículo se dan bases teóricas para modelar funciones distancia sobre el espacio euclidiano usual Rⁿ, donde distancia se puede referir a kilómetros, litros de combustible consumidos, tiempo de recorrido, o costo de transporte. En nuestro enfoque, una función distancia d se obtiene a partir de una función Fo llamada la función fundamental de d. Estas funciones distancia, a diferencia de las métricas, pueden ser asimétricas y no positivas definidas, y a diferencia de las normas Lp, pueden ser no uniformes. Se ilustra el marco teórico propuesto a través del modelado de una función distancia asimétrica y no uniforme sobre R² la cual puede tomar valores negativos.

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