Bounds on Area Involving Lattice Size

Abstract

The lattice size of a lattice polygon $P$ is a combinatorial invariant of $P$ that was recently introduced in relation to the problem of bounding the total degree and the bi-degree of the defining equation of an algebraic curve. In this paper, we establish sharp lower bounds on the area of plane convex bodies $P\subset\mathbb{R}^2$ that involve the lattice size of $P$. In particular, we improve bounds given by Arnold, and Bárány and Pach. We also provide a classification of minimal lattice polygons $P\subset\mathbb{R}^2$ of fixed lattice size ${\operatorname{ls_\square}}(P)$.