Characterizing Certain Staircase Convex Sets in R

Let C = {C1, . . . , Cn} be a family of distinct boxes in R whose intersection graph is a tree, and let S = C1∪· · ·∪Cn. Let T ⊆ S. The set T lies in a staircase convex subset of S if and only if for every a, b in T there is an a−b staircase path in S. This result, in turn, yields necessary and sufficient conditions for S to be a union of k staircase convex sets, k ≥ 1. Analogous results characterize S as a union of k staircase starshaped sets. Further, when d ≥ 3, the set S above will be staircase convex if and only if for every chain A of boxes in C, each projection of A into a coordinate hyperplane is staircase convex. Finally, if S is any orthogonal polytope in R, d ≥ 2, S is staircase convex if and only if, for every j-flat F parallel to a coordinate flat, F ∩ S is connected, 1 ≤ j ≤ d− 1. MSC 2000: 52.A30, 52.A35