A note on the length, measured along fixed orientations, of a path connecting any two points relative to their straight line distance

摘要

Let D= (d_1,...,d_n)∈ R~(n×n) be an invertible matrix and L_1 (x) denote the well-known 1-norm of x ∈ R~n. In this note we analyse the kth moment of the ratio L_1 (D~1z) and its asymptotics in special cases, where z is uniformly distributed over the unit hypersphere. Conceptually, L_1 (D~1 z) can be seen as the (weighted) length, measured along fixed orientations, of a path connecting any two points relative to their straight line distance.