Generalizations of Scott's inequality and Pick's formula to rational polygons

TL;DR

This paper extends classical lattice polygon inequalities to rational polygons, providing sharp bounds on boundary points and area based on interior points and denominators, with applications to Ehrhart quasipolynomials.

Contribution

It generalizes Scott's inequality and Pick's formula to rational polygons, offering new bounds and characterizations for these geometric quantities.

Findings

Sharp upper bound on boundary lattice points in terms of interior points and denominator

Sharp bounds on area based on interior points, boundary points, and denominator

Characterization of minimizers and maximizers for these bounds

Abstract

We prove a sharp upper bound on the number of boundary lattice points of a rational polygon in terms of its denominator and the number of interior lattice points, generalizing Scott's inequality. We then give sharp lower and upper bounds on the area in terms of the denominator, the number of interior lattice points, and the number of boundary lattice points, which can be seen as a generalization of Pick's formula. Minimizers and maximizers are described in detail. As an application, we derive bounds for the coefficients of Ehrhart quasipolymials of half-integral polygons.