Intersection patterns of convex sets

Let K ₁,...K_n 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=...=K_r=R^d and K _r+1,...,K_n 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.