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Partitions of Points into Simplices withk-dimensional Intersection. Part I: The Conic Tverberg’s Theorem

✍ Scribed by Jean-Pierre Roudneff

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
148 KB
Volume
22
Category
Article
ISSN
0195-6698

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✦ Synopsis

Tverberg's 1966 theorem asserts that every set X of (m -1)(d + 1) + 1 points in R d has a partition X 1 , X 2 , . . . , X m such that m i=1 conv X i = φ. We give a short and elementary proof of a theorem on convex cones which generalizes this result. As a consequence, we deduce several divisibility properties, including the characterization of extremal sets which have no partition such that m i=1 conv X i is at least one-dimensional and, in the particular cases m = 3 and m = 4, the proof of Reay's conjecture that every set of (m -1)(d + 1) + k + 1 points in general position in R d has a partition such that m i=1 conv X i is at least k-dimensional.

📜 SIMILAR VOLUMES

Partitions of Points into Simplices with
Partitions of Points into Simplices withk-dimensional Intersection. Part II: Proof of Reay’s Conjecture in Dimensions 4 and 5
✍ Jean-Pierre Roudneff 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 236 KB

A longstanding conjecture of Reay asserts that every set X of (m-1)(d +1)+k+1 points in general position in R d has a partition X 1 , X 2 , . . . , X m such that m i=1 conv X i is at least k-dimensional. Using the tools developed in [13] and oriented matroid theory, we prove this conjecture for d =