Determining regular networks for each given lp function (p ∈ [1, 2])

The problem of how to decide which distance function should be fitted to actual road distances has been studied during recent years and has special importance, among others, in location problems. For this purpose lp norms have been used. However, when a new transportation network is going to be designed the, in some way opposite, problem arises of how to connect regularly the given points so that network distances can be approximated by a given lp function. Since the problem has an easy solution for p = 1 and p = 2, and as for p > 2 it has no sense, we studied the case in which p ∈ (1, 2). For each value of p in this interval there exists an angle φ such that the block norm corresponding to a dense network in which the trips only have the directions given by the vectors g1 = (1, 0), g2 = (cos φ, sin φ), −g1, −g2, is used to determine the regular network for which distances can be approximated by using the given lp norm.