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

Abstract

Let $${\mathcal {K}}$$ be a finite family of orthogonal polytopes in $${\mathbb {R}}^{d}$$ such that, for every nonempty subfamily $${\mathcal {K}}^{\prime }$$ of $${\mathcal {K}}, \cap \{K: K$$ in $$ {\mathcal {K}}^{\prime }\}$$ , if nonempty, is a finite union of boxes whose intersection graph is a tree. Let k and n be fixed integers, $$1 \le k\le 2^{n}$$ . Assume that certain visibility sets have connected intersections and that, for every $$2 (d+1)^{n+1}$$ (or fewer) member subset $${\mathcal {K}}^{\prime \prime }$$ of $${\mathcal {K}}, \cap \{K : K$$ in $$ {\mathcal {K}}^{\prime \prime } \}$$ is nonempty and starshaped via k-staircase paths. Then $$\cap \{K : K$$ in $$ {\mathcal {K}}\}$$ is nonempty and starshaped via k-staircase paths as well.