d-Representability of simplicial complexes of fixed dimension

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Let K be a simplicial complex with vertex set V = v₁,..., vₙ. The complex K is d-representable if there is a collection C₁,...,Cₙ of convex sets in Rᵈ such that a subcollection Cᵢ₁,...,Cᵢⱼ has a nonempty intersection if and only if vᵢ₁,...,vᵢⱼ is a face of K. In 1967 Wegner proved that every simplicial complex of dimension d is (2d+1)-representable. He also suggested that his bound is the best possible, i.e., that there are d-dimensional simplicial complexes which are not 2d-representable. However, he was not able to prove his suggestion. We prove that his suggestion was indeed right. Thus we add another piece to the puzzle of intersection patterns of convex sets in Euclidean space.