Note on the integer geometry of bitwise XOR

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.