Characterization of f-vectors of families of convex sets in R ^d Part I: Necessity of Eckhoff's conditions

Let K=K ₁,...,K_n be a family of n convex sets in R ^d . For 0≦i<n denote by f _i the number of subfamilies of K of size i+1 with non-empty intersection. The vector f(K) is called the f-vectors of K. In 1973 Eckhoff proposed a characterization of the set of f-vectors of finite families of convex sets in R ^d by a system of inequalities. Here we prove the necessity of Eckhoff's inequalities. The proof uses exterior algebra techniques. We introduce a notion of generalized homology groups for simplicial complexes. These groups play a crucial role in the proof, and may be of some independent interest.