Let F be a family of n+1 convex sets in R^d, each n of which have a point in common, such that F is starshaped. If either all members of F are closed or all members of F are open, then the intersection of F is nonempty. This result which strengthens a theorem by V.Klee follows from a topological theorem of C.D.Horvath and M.Lassonde. We present a simple geometric proof in the spirit of Klee''s proof. This immediately provides an alternative proof of a Helly type theorem due to M.Breen. An abstract vector space variant of the above result is given, too.
A simple geometric proof of a theorem for starshaped unions of convex sets / L. Vesely. - In: ACTA UNIVERSITATIS CAROLINAE. MATHEMATICA ET PHYSICA. - ISSN 0001-7140. - 49:(2008), pp. 79-82.
A simple geometric proof of a theorem for starshaped unions of convex sets
L. VeselyPrimo
2008
Abstract
Let F be a family of n+1 convex sets in R^d, each n of which have a point in common, such that F is starshaped. If either all members of F are closed or all members of F are open, then the intersection of F is nonempty. This result which strengthens a theorem by V.Klee follows from a topological theorem of C.D.Horvath and M.Lassonde. We present a simple geometric proof in the spirit of Klee''s proof. This immediately provides an alternative proof of a Helly type theorem due to M.Breen. An abstract vector space variant of the above result is given, too.Pubblicazioni consigliate
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