Many authors, e.g., Rosenfeld and Pfaltz, Borgefors..., have proposed efficient and/or accurate approximations of euclidian distance on a 2D or 3D grid with methods which are connected, more or less directly, to norm derived distances, e.g., with Lp norms. This paper enlarges the scope in a continuous and m-dimensional framework. It presents a new broad class of distances, called 'sandwich' or 'periodic' distances. They are obtained by compounding in a periodic manner a certain number of norm-derived distances. The main result of this paper is the proof of a sufficient condition under which the triangular inequality is fulfilled, i.e., that the unit balls of the compounded distances belong to an ascending chain. Moreover, the theory includes weighted distances, giving this tool a high degree of flexibility.