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) ≥ 12 · 6n / 2(n + 1) − n + 12n!. Also limn → ∞n [H(n)]1 / n ≈ 0.9281. Explicit bounds for T(n) are tabulated for n ≤ 10, and we mention some other results on hyperbolic volumes.