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

Let \(\mathcal{C}\) be a finite family of boxes in \(\mathbb{R}^d\), \(d \geq 3\), 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, \(1 \leq n \leq d-2\), an analogous result holds for composites of \(n\) coordinate projections of \(D\) into \((d-n)\)-dimensional flats.