A Lower Bound for the Simplexity of then-Cube via Hyperbolic Volumes

Let T(n) denote the number of n-simplices in a minimum cardinality decomposition of the n-cube into n-simplices. For n 1 we show that T(n) H(n), where H(n) is the ratio of the hyperbolic volume of the ideal cube to the ideal regular simplex. H(n) 1 2 6 n=2 (n+1) ? n+1 2 n!. Also lim n!1 p nH(n)] 1=n 0:9281. Explicit bounds for T(n) are tabulated for n 10, and we mention some other results on hyperbolic volumes.