2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/124507We 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.AMS-LaTeX, 19 pages; paper reorganized, introduction added, bibliography updated; typos correctedAlgebraic GeometryCombinatoricsNumber Theory14M25, 14J45; 52B20; 11D75, 11H06text