Journal of Geometry Vol. 38 (1990) 0047-2468/90/020023-1651.50+0.20/0 (c) 1990 Birkh~user Verlag, Basel THE DISCRETE VERSION OF A GEOMETRIC DUALITY THEOREM Dedicated to Professor Srinivas Ramanujan on his birth centenary Partha Pratim Das It was shown by Rhodes [1] that a theorem about subsets in the plane specified by the Euclidean metric generalizes to an interesting duality between the absolute and the maximum metrics in the real plane. In this paper the discrete version of this duality is shown to hold between the cityblocM (absolute) and the chessboard (maximum) metrics in the quantized space. The characterization of the "bisector" and the "near-bisector' under the above metrics is obtained as a by-product. I. INTRODUCTION Let R be the set of real numbers and R + be the set of non- negative real numbers. Hence R 2= (~= (Xl,X 2) : Xl,X 2 8 R} is the real plane. If d : R 2 x R 2 ---> R + is a metric (i.e., d is total, positive definite, symmetric and trianglar) on R 2 then we can define the following quantities in terms of d. (A) Sgt of .Disk Points: D(E,r) = {~ : ~ 8 R 2, d(~,~) _< r and r e R +} where [ ~ p2 is the centre and r is the radius. (B) Sgt o( Between points: B(K, Z) = {~ : ~ 8 R 2, d(K,~)+d(~,Z) = d(K,Z)}; ~,Z 8 R 2. From the triangularity of d, B is clearly the set of points lying on some minimal path from ~ to ~ as defined by the metric d. (C) Set of Equidistant (Bisector) Points: E(x__,y_) = {~ : ~ @ R2, d(]s = d(y_,u_)}; x_,y_ 8 R 2.