## Misunderstanding geographical distances: two errors and an issue in the interpretation of violations of triangle inequality

#### L'HOSTIS

Type de document

ARTICLE A COMITE DE LECTURE REPERTORIE DANS BDI (ACL)

Langue

anglais

Auteur

L'HOSTIS

Résumé / Abstract

We investigate the meaning of the mathematical properties of distances in the fields of geography and economy. The key property for spatiality is the triangle inequality (TI) as ensurring the optimality of distance. We identify three different situations where several authors identify violations of the TI. We consider all of them as errors of interpretation. The first error consists in considering sub-optimal measurements as distances. Yet distances are necessarily optimal since they respect the TI. The second set of error, which is the most widespread, involves a confusion between the Euclidean straight line and the minimum path. The errors consist in considering the presence of a detour as a violation of the TI, while this situation simply corresponds to a non-Euclidean distance. The third error concerns the issue of additivity of distances. The commonplace situation in geographical space where a break is needed to provide the energy necessary to renew the movement, is considered by some authors as another violation of the TI. We argue that as these routes are optimal, the TI must hold. We finally introduce a distance function that allows considering sub-additivity and over-additivity of distances, and in the same time respect the TI.

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