Characterization of f-vectors of families of convex sets in Rd part II: Sufficiency of Eckhoff's conditions

Gil Kalai

doi:10.1016/0097-3165(86)90079-8

Characterization of f-vectors of families of convex sets in R^d part II: Sufficiency of Eckhoff's conditions

Gil Kalai^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

22 Scopus citations

Abstract

Let K = {K₁,..., 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) = (f₀, f₁,...) is called the f-vector 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. In part I we proved the necessity of Eckhoffs inequalities and here we prove their sufficiency.

Original language	English
Pages (from-to)	167-188
Number of pages	22
Journal	Journal of Combinatorial Theory. Series A
Volume	41
Issue number	2
DOIs	https://doi.org/10.1016/0097-3165(86)90079-8
State	Published - Mar 1986
Externally published	Yes

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abstract = "Let K = \{K1,..., Kn\} be a family of n convex sets in Rd. For 0≤ii the number of subfamilies of K of size i + 1 with non-empty intersection. The vector f(K) = (f0, f1,...) is called the f-vector of K. In 1973, Eckhoff proposed a characterization of the set of f-vectors of finite families of convex sets in Rd by a system of inequalities. In part I we proved the necessity of Eckhoffs inequalities and here we prove their sufficiency.",

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Characterization of f-vectors of families of convex sets in R^d part II: Sufficiency of Eckhoff's conditions

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