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

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


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