A lower bound for the simplexity of the \(n\)-cube via hyperbolic volumes. (English) Zbl 0982.51015
Summary: Let \(T(n)\) denote the number of \(n\)-simplices in a minimum cardinality decomposition of the \(n\)-cube into \(n\)-simplices. For \(n\geq 1\), we show that \(T(n)\geq H(n)\), where \(H(n)\) is the ratio of the hyperbolic volume of the ideal cube to the ideal regular simplex. \(H(n)\geq\frac 12\cdot 6^{n/2}(n+1)^{-\frac{n+1}{2}} n!\). Also \(\lim_{n\to\infty}\sqrt{n}[H(n)]^{1/n}\approx 0.9281\). Explicit bounds for \(T(n)\) are tabulated for \(n\leq 10\), and we mention some other results on hyperbolic volumes.
MSC:
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
| 52B11 | \(n\)-dimensional polytopes |
| 52B55 | Computational aspects related to convexity |
Online Encyclopedia of Integer Sequences:
Simplexity of the n-cube: minimal cardinality of triangulation of n-cube using n-simplices whose vertices are vertices of the n-cube.References:
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