A Helly-type theorem for intersections of certain orthogonal polytopes starshaped via \(k\)-staircase paths (Q2204387)

For a finite family \(\mathcal{K}\) of orthogonal polygons in \(\mathbb{R}^d\) and for fixed \(n \geq 1\), the property that appropriate visibility sets have connected intersections for \(\mathcal{K}\) and \(n\) is introduced. It allows the author to extend the Helly-type theorems given by \textit{N. A. Bobylev} [J. Math. Sci., New York 105, No. 2, 1819--1825 (2001; Zbl 1013.52006); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat Obz. 68, 16--26 (1999)] and \textit{M. Breen} [Period. Math. Hung. 68, No. 1, 45--53 (2014; Zbl 1299.52010)], by proving the following result: Let \(\mathcal{K}\) be a finite family of orthogonal polytopes in \(\mathbb{R}^d\) such that, for every nonempty subfamily \(\mathcal{K}'\) of \(\mathcal{K}\), \(\bigcap\{K : K \in \mathcal{K}'\}\), if nonempty, is a finite union of boxes whose intersection graph is a tree. Let \(k\) and \(n\) be fixed integers, \(0\leq n\), \(1 \leq k \leq 2n\). Assume that, for every \(2(d + 1)^{n+1}\) (or fewer) member subset \(\mathcal{K}''\) of \(\mathcal{K}\), \(\bigcap\{K : K \in \mathcal{K}''\}\) is nonempty and starshaped via \(k\)-staircase paths. When \(1 \leq n\) and \(2 \leq k\), assume that appropriate visibility sets have connected intersections for \(\mathcal{K}\) and \(n\), and assume that \(k\)-visibility sets for \(S\) have connected intersections. Then \(\bigcap\{K : K \in \mathcal{K}\} \equiv S\) is nonempty and starshaped via \(k\)-staircase paths as well.