Bounds on area involving lattice size (Q6117269)

Summary: 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 \(\operatorname{ls}_{\square}(P)\).