Estimation of errors between Euclidean and m-neighbor distance (Q1262140)

In n-dimensional grid point space, distance functions \(d_{n,m}\) are defined (called the m-neighbor distance) which may be used to approximate the Euclidean distance E. The definition of the m-neighbor distance \(d_{n,m}\) is based on a special neighborhood of grid points in n- dimensional grid point space, and n different neighborhoods are considered. The properties of approximation errors between \(d_{n,m}\) and E are dealt with. It is proved, that the proportional error (the ratio between \(d_{n,m}\) and E) is bounded. Using error measures based on this proportional error a method is proposed for selecting that m which gives ``the least error to approximate \(E''\).