Note on the integer geometry of bitwise XOR

We consider the set N of non-negative integers together with a distance d defined as follows: given two integers x,y∈N, d(x,y) is, in binary notation, the result of performing, digit by digit, the “XOR” operation on (the binary notations of) x and y. Dawson, in Combinatorial Mathematics VIII, Geelong, 1980, Lecture Notes in Mathematics, 884 (1981) 136, considers this geometry and suggests the following construction: given k different integers x1,…,xk∈N, let Vi be the set of integers closer to xi than to any xj with j≠i, for i,j=1,…,k. Let V=(V1,…,Vk) and X=(x1,…,xk). V is a partition of {0,1,…,2n−1} which, in general, does not determine X. s paper, we characterize the convex sets of this geometry: they are exactly the line segments. Given X and the partition V determined by X, we also characterize in easy terms the ordered sets Y=(y1,…,yk) that determine the same partition V. This, in particular, extends one of the main results of Combinatorial Mathematics VIII, Geelong, 1980, Lecture Notes in Mathematics, 884 (1981) 136.