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.In this 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.