Hyperspheres of weighted distances in arbitrary dimension

In a previously reported work, a distance function was proposed which defines the distance between any pair of points as the weighted sum of their ordered coordinate differences. We call this distance function in this work as linear combination form of weighted distance (LWD), and observe that if an LWD is a norm, it can be expressed in an equivalent form, which is associated with a chamfering mask. We refer to this class of distance functions as chamfering weighted distances (CWD). In this work, properties of hyperspheres of CWDs in arbitrary dimension are discussed. We have derived expressions for the vertices, surface areas and volumes of n-D hyperspheres. These are used in defining geometric error measures to study the proximity of these distance functions to Euclidean metrics. We have also used other analytical error measures to consider their suitability in approximating Euclidean distances.

► Computing vertices, volumes, and surface areas of hyperspheres of that class of weighted distances in arbitrary dimension.
► Finding good approximation of Euclidean norm among that class of weighted distance functions using geometric and analytical error measures.