Lattice Size and Area of Plane Convex Bodies

Abstract

The lattice size of a lattice polygon P was introduced and studied by Schicho, and by Castryck and Cools 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 ⊂ R 2 that involve the lattice size of P . In particular, we improve bounds established by Arnold, and Bárány and Pach. We also provide a classification of minimal lattice polygons P ⊂ R 2 of fixed lattice size ls (P ).