On the metrics of Chaudhuri, Murthy and Chaudhuri

Abstract The paper considers the approximation of Euclidean distance in n-dimensional space by linear combinations of the L1 and L∞ metrics. Maximal proportional errors for the one parameter family introduced by Chaudhuri, Murthy and Chaudhuri are calculated. Estimates of the optimal parameters for one parameter families are obtained by solving a quartic equation numerically. The maximal proportional errors for these parameters are much smaller than those for the parameters chosen by Chaudhuri et al. It is shown that for two parameter families the corresponding quartic equation can be solved algebraically. Thus the behaviour of the optimal solutions can be seen more clearly, though the approximations to the Euclidean metric are not substantially improved.