The Helly theorem for star-shaped sets
The main result of this paper is the following analogue to Helly's theorem. Let \((K_{i})_{i\in I}\) be a family of star-shaped compact sets in Euclidean \(N\)-space \(E^{N}\), and let \(\operatorname {card}I\geq N+1\). If for every \(k\leq N+1\) the intersection of any \(k\) sets from this family is nonempty and star-shaped, then the intersection \(\bigcap_{i\in I}K_{i}\) is nonempty and star-shaped. There is also a dual theorem concerning the union instead of the intersection. Finally, the following theorem is proved. Let \((C_{i})_{i\in I}\) be a finite family of convex compact sets in \(E^{N}\) with \(\operatorname {card}I\geq N+1\). If for every \(k\leq N+1\) the union of any \(k\) sets from this family is star-shaped, the intersection \(\bigcap_{i\in I}C_{i}\) is nonempty.
scientific article; zbMATH DE number 1821778