Israel Journal of Mathematics

, Volume 48, Issue 2, pp 161–174

Intersection patterns of convex sets

  • Gil Kalai
Article

DOI: 10.1007/BF02761162

Cite this article as:
Kalai, G. Israel J. Math. (1984) 48: 161. doi:10.1007/BF02761162

Abstract

LetK1,…Kn be convex sets inRd. For 0≦i<n denote byfithe number of subsetsS of {1,2,…,n} of cardinalityi+1 that satisfy ∩{Ki∶i∈S}≠Ø. We prove:Theorem.If fd+r=0 for somer r>=0, then {fx161-1} This inequality was conjectured by Katchalski and Perles. Equality holds, e.g., ifK1=…=Kr=Rd andKr+1,…,Kn aren−r hyperplanes in general position inRd. The proof uses multilinear techniques (exterior algebra). Applications to convexity and to extremal set theory are given.

Copyright information

© The Weizmann Sciene Press of Israel 1984

Authors and Affiliations

  • Gil Kalai
    • 1
  1. 1.Institute of Mathematics The Hebrew University of JerusalemJerusalemIsrael