Kołodziejczyk, Krzysztof On starshapedness of the union of closed sets in \({\mathbb{R}}^ n\). (English) Zbl 0635.52006 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 Cited in 5 Documents 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 PDF BibTeX XML Cite \textit{K. Kołodziejczyk}, Colloq. Math. 53, No. 1--2, 193--197 (1987; Zbl 0635.52006) Full Text: DOI