On approximating Euclidean metrics by weighted t-cost distances in arbitrary dimension

In this work, we propose a new class of distance functions called weighted t-cost distances. This function maximizes the weighted contribution of different t-cost norms in n-dimensional space. With proper weight assignment, this class of function also generalizes m-neighbor and octagonal distances. A non-strict upper bound (denoted as Ru in this work) of its relative error with respect to Euclidean norm is derived and an optimal weight assignment by minimizing Ru is obtained. However, it is observed that the strict upper bound of weighted t-cost norm may be significantly lower than Ru. For example, an inverse square root weight assignment leads to a good approximation of Euclidean norm in arbitrary dimension.

Research highlights
► The present work proposes a new family of distance functions names as weighted t-cost distances, whose property of metricity is proved.
► The concept of weighted t-cost distances generalizes different digital distances reported earlier, namely, m-neighbor distances, t-cost distances and octagonal distances.
► A sub-class of weighted t-cost distances, named as inverse square root weighted t-cost distances is found to be good approximator of Euclidean norms in arbitrary dimension. Strict upper bounds of their relative errors with respect to Euclidean norms are also presented.