$d$-representability of simplicial complexes of fixed dimension

• Main Author: Martin Tancer
• Format: Article
• Language: English
• Published: Carleton University 2011-11-01
• Series: Journal of Computational Geometry
• Online Access: http://jocg.org/index.php/jocg/article/view/73

<p>Let <strong>K</strong> be a simplicial complex with vertex set <em>V</em> = <em>v</em>₁,…,<em>v</em>_<em>n</em>. The complex <strong>K</strong> is <em>d</em>-representable if there is a collection {<em>C</em>₁,…,<em>C</em>_<em>n</em>} of convex sets in <strong>R</strong>^<em>d</em> such that a subcollection {<em>C</em>_{<em>i</em>₁},…,<em>C</em>_{<em>i</em>_<em>j</em>}} has a nonempty intersection if and only if {<em>v</em>_{<em>i</em>₁},…,<em>v</em>_{<em>i</em>_<em>j</em>}} is a face of <strong>K</strong>.<br /><br />In 1967 Wegner proved that every simplicial complex of dimension <em>d</em> is (2<em>d</em>+1)-representable. He also conjectured that his bound is the best possible, i.e., that there are <em>d</em>-dimensional simplicial complexes which are not 2<em>d</em>-representable. However, he was not able to prove his conjecture.<br /><br />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.</p>