Volume and lattice points of reflexive simplices

We prove sharp upper bounds on the volume and the number of lattice points on edges of higher-dimensional reflexive simplices. These convex-geometric results are derived from new number-theoretic bounds on the denominators of unit fractions summing up to one. The main algebro-geometric application is a sharp upper bound on the anticanonical degree of higher-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number one, where we completely characterize the case of equality.Comment: AMS-LaTeX, 19 pages; paper reorganized, introduction added, bibliography updated; typos correcte

Lattice Delone simplices with super-exponential volume

In this short note we give a construction of an infinite series of Delone simplices whose relative volume grows super-exponentially with their dimension. This dramatically improves the previous best lower bound, which was linear.Comment: 7 pages; v2: revised version improves our exponential lower bound to a super-exponential on

Lower bounds for the simplexity of the n-cube

In this paper we prove a new asymptotic lower bound for the minimal number of simplices in simplicial dissections of $n$-dimensional cubes. In particular we show that the number of simplices in dissections of $n$-cubes without additional vertices is at least $(n+1)^{\frac {n-1} 2}$.Comment: 10 page

The Hyperdeterminant and Triangulations of the 4-Cube

The hyperdeterminant of format 2 x 2 x 2 x 2 is a polynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4-cube. The 87959448 regular triangulations of the 4-cube are classified into 25448 D-equivalence classes, one for each vertex of the Newton polytope. The 4-cube has 80876 coarsest regular subdivisions, one for each facet of the secondary polytope, but only 268 of them come from the hyperdeterminant.Comment: 30 pages, 6 figures; An author's name changed, typos fixe