An Enlarged Class of Orthogonal Polytopes Having Staircase Convex Kernels

Let \(\mathcal{C}\) be a finite family of distinct boxes in \(\mathbb{R}^d\), with \(G\) the intersection graph of \(\mathcal{C}\), and let \(S = \cup\{C : C \in \mathcal{C}\}\). For each block of \(G\), assume that the corresponding members of \(\mathcal{C}\) have a staircase convex union. Then when \(S\) is staircase starshaped, its staircase kernel will be a staircase convex set. Moreover, this result (and others) will hold for more general families \(\mathcal{C}\) as well.