Using Chains of Boxes to Recognize Staircase Starshaped Sets in ${\mathbb{R}}^d$

Abstract

Let $\mathcal{C}$ be a finite family of boxes in ${\mathbb{R}}^d$, $d\ge 3$, with $S=\cup \{C:C\in \mathcal{C}\}$ connected and $p\in S$. Assume that, for every geodesic chain $D$ of $\mathcal{C}$-boxes containing $p$, each coordinate projection $\pi (D)$ of $D$ is staircase starshaped with $\pi (p)\in \text{Ker}\pi (D)$. Then $S$ is staircase starshaped and $p\in \text{Ker}S$. For $n$ fixed, $1\le n\le d-2$, an analogous result holds for composites of $n$ coordinate projections of $D$ into $(d-n)$-dimensional flats.