Partitions of Points into Simplices withk-dimensional Intersection. Part II: Proof of Reay’s Conjecture in Dimensions 4 and 5
✍ Scribed by Jean-Pierre Roudneff
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 236 KB
- Volume
- 22
- Category
- Article
- ISSN
- 0195-6698
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✦ Synopsis
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 = 4 and d = 5. How about, to that end, we introduce the notion of a k-lopsided oriented matroid and we characterize these combinatorial objects for certain values of k. Divisibility properties for subsets of R d with other independence conditions are also obtained, thus settling several particular cases of a generalization of Reay's conjecture.