Intersection patterns of convex sets

Gil Kalai*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

85 Scopus citations

Abstract

Let K 1,...Kn be convex sets in R d. For 0≦i<n denote by f ithe number of subsets S of {1,2,..., n} of cardinality i+1 that satisfy ∩{K i:i∈S}≠Ø. We prove:Theorem. If f d+r=0 for somer r>=0, then {fx161-1} This inequality was conjectured by Katchalski and Perles. Equality holds, e.g., if K 1=...=Kr=Rd and K r+1,...,Kn are n-r hyperplanes in general position in R d. The proof uses multilinear techniques (exterior algebra). Applications to convexity and to extremal set theory are given.

Original languageEnglish
Pages (from-to)161-174
Number of pages14
JournalIsrael Journal of Mathematics
Volume48
Issue number2-3
DOIs
StatePublished - Jun 1984

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