An Enlarged Class of Orthogonal Polytopes Having Staircase Convex Kernels

Abstract

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.