A Helly theorem for an enlarged class of orthogonally starshaped sets in $$\mathbb {R}^d$$ R d

Abstract

Consider a finite family $$\mathcal {C}$$ of distinct boxes in $$\mathbb {R}^d$$ , with G the intersection graph of $$\mathcal {C}$$ and with $$S \equiv \cup \{C : C \, \hbox {in} \, \mathcal {C}\}$$ . Assume that G is connected and, for each block B of G, assume that the corresponding members of $$\mathcal {C}$$ have a staircase convex union D(B). When this occurs, we say that the orthogonal polytope S has suitable properties. Now let $$\mathbf{S}$$ be a finite family of orthogonal polytopes such that, for every nonempty subfamily $$\mathbf{S}^{'}$$ of $$\mathbf{S}$$ , the corresponding intersection $$S^{'} \equiv \cap \{S : S \, \hbox {in} \, \mathbf{S}^{'} \}$$ (if nonempty) has suitable properties and preserves transitions between certain D(B) sets. If every $$d +1$$ (not necessarily distinct) members of $$\mathbf{S}$$ meet in a (nonempty) staircase starshaped set, then $$S_0 \equiv \cap \{S : S \, \hbox {in} \, \mathbf{S} \}$$ is nonempty and staircase starshaped as well.