  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.
