Intersections of Sets Expressible as Unions of \(k\) Star-Shaped Sets

Abstract

Let \(\mathcal{K}\) be a family of sets in \(\mathbb{R}^d\) and let \(k\) be a fixed natural number. Assume that every countable subfamily of \(K\) has an intersection expressible as a union of \(k\) starshaped sets, each having a \(d\)-dimensional kernel. Then \(S = \cap \{K : K \in \mathcal{K}\}\) is nonempty and is expressible as a union of \(k\) such starshaped sets.

If members of \(K\) are compact and every finite subfamily of \(\mathcal{K}\) has as its intersection a union of \(k\) starshaped sets, then \(S\) again is a union of \(k\) starshaped sets. An analogous result holds for unions of \(k\) convex sets. Finally, dual results hold for unions of subfamilies of \(\mathcal{K}\).