On starshapedness of the union of closed sets in \({\mathbb{R}}^ n\). (English) Zbl 0635.52006

Kołodziejczyk, Krzysztof

Colloq. Math. 53, No. 1-2, 193-197 (1987).

It is shown that Helly’s Theorem is equivalent to the following: Let F be a finite family of closed sets in \({\mathbb{R}}^ n;\) if any \(n+1\) members of F has a starshaped union, then ker(\(\cup F)\neq \emptyset\). The kernel of a set consists of the points that can ‘see’ all points of the set. Using a theorem of Katchalski, a lower bound for the dimension of ker(\(\cup F)\) is determined.

Reviewer: G.Sierksma

MSC:

52A35	Helly-type theorems and geometric transversal theory
52A30	Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)

Keywords:

Krasnosel’skij’s theorem; Helly’s Theorem; starshaped union