TY - JOUR
T1 - Intersection patterns of convex sets
AU - Kalai, Gil
PY - 1984/6
Y1 - 1984/6
N2 - Let K 1,...Kn be convex sets in R d. For 0≦i 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.
AB - Let K 1,...Kn be convex sets in R d. For 0≦i 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.
UR - http://www.scopus.com/inward/record.url?scp=51249182570&partnerID=8YFLogxK
U2 - 10.1007/BF02761162
DO - 10.1007/BF02761162
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AN - SCOPUS:51249182570
SN - 0021-2172
VL - 48
SP - 161
EP - 174
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 2-3
ER -