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Empty simplices in Euclidean space. (English) Zbl 0639.52006
Let P be a set of n points in \({\mathbb{R}}^ d\) in general position. A simplex spanned by \(d+1\) distinct points of P is called empty if it contains no point of P in its interior. Let \(f_ d(P)\) denote the number of empty simplices in P. Trivially, \(f_ d(P)\leq \left( \begin{matrix} n\\ d+1\end{matrix} \right)\), with equality if P is the vertex set of a convex d-polytope. By a result of Katchalski and Meir, \(f_ d(P)\geq \left( \begin{matrix} n-1\\ d\end{matrix} \right).\)
The authors describe a random construction for P which proves \(f_ d(P)<K(d)\left( \begin{matrix} n\\ d\end{matrix} \right)\), with K(d) a constant depending only on d. Several related questions are discussed.
Reviewer: E.Schulte

52A37 Other problems of combinatorial convexity
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
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