All geographical distances are optimal

Triangular inequality is one of the four mathematical properties of distance. Its respect derives from the optimal nature of the measurement of distance. This demonstration (L'Hostis 2016, 2017) reveals key aspects of distances and geographical spaces. We develop this argument by investigating the idea of the optimality of distance through a mathematical and geometric discussion, and by dealing with empirical approaches to applied geography. The first part of the paper explores the consequences of considering that the mathematical property of triangle inequality is always respected. In fact, no violations of triangle inequality are observed in geographical spaces. The study of the optimality of distances in empirical approaches confirms the key role of the property of triangle inequality. The general principle of least-effort applies for most movements and spacings. In addition, however, trajectories with multiple detours, like those of shoppers, runners and nomads, are optimal from a certain point of view. This is also the case for excess travel, i.e. a situation of disjunction between an optimum as perceived by a person in movement and an optimum as perceived by an external observer. Any movement, any spacing within and between cities and in geographical space in general, exhibits a form of optimality, and all geographical distances are optimal.