On combinatorial problem concerning partitions of a box into boxes

Abstract

We consider partitions of n-dimensional boxes in R^n, n>1, into a finite number of boxes with pairwise disjoint interiors. We study sets X \subseteq (0,\infty) with the Property (W_n): for every n-dimensional box P and every partition of P, if each constituent box has one side with the length belonging to X, then the length of one side of P belongs to X. We prove that the set X \subseteq (0,\infty) has Property (W_n) if and only if X is closed with respect to the operations: x+y and x+y+z-2min(x,y,z).