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

Let \( \mathcal{C} = \{C_1, \ldots, C_n\} \) 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.