A Note Concerning Kernels of Staircase Starshaped Sets in ${\mathbb{R}}^d$

Abstract

Let $\mathcal{C}=\{C_1,\dots ,C_n\}$ be a family of distinct boxes in ${\mathbb{R}}^d$, and let $S=C_1\cup \cdots \cup C_n$. Assume that $S$ is staircase starshaped. If the intersection graph of $\mathcal{C}$ is a tree, then the staircase kernel of $S$, $\ker S$, will be staircase convex. However, an example in ${\mathbb{R}}^3$ reveals that, without this requirement on the intersection graph of $\mathcal{C}$, components of $\ker S$ need not be staircase convex. Thus the structure of the kernel in higher dimensional staircase starshaped sets provides a striking contrast to its structure in planar sets.