Coefficients of the solid angle and Ehrhart quasi-polynomials

Fabrício Caluza Machado, Sinai Robins

Published 2019-12-17Version 1

Given a rational polytope $P$, we give a local formula for the codimension two quasi-coefficient of the solid angle sum of $P$, which is one version of a discretized volume of $P$. This local formula is valid for all positive real dilates of a rational polytope in $\mathbb R^d$. As a consequence, we show that the classical Ehrhart polynomial, which is another discretized volume of $P$, also has a similar local formula for the codimension one and codimension two quasi-coefficients. The formula for the codimension one quasi-coefficient extends the classical interpretation of half the sum of the relative volumes of the facets, valid for integer dilations, to a formula valid for real dilations. Some of the present methods offer a further development for the initial approach of Diaz, Le, and Robins.