Estimation of errors between Euclidean and m-neighbor distance

In this paper the absolute and relative errors between Euclidean and m-neighbor distance have been studied for n-D digital geometry. Over the n-D space the absolute error has been shown to be unbounded. However, the relative error is shown to be bounded from above by nm. An efficient algorithm for determining the maxima of the relative error is presented. The existence of a still simpler and more efficient algorithm is conjectured. The use of the estimate of maximum relative error as a measure of the quality of approximation of Euclidean distance by an m-neighbor distance is discussed. The nature of the error function is shown graphically.