Bounds on Area Involving Lattice Size

TL;DR

This paper establishes sharp lower bounds on the area of convex bodies in the plane based on their lattice size, improving previous bounds and classifying minimal polygons of fixed lattice size.

Contribution

It provides new sharp lower bounds on area involving lattice size and classifies minimal lattice polygons for fixed lattice size.

Findings

Improved bounds on area involving lattice size.

Classification of minimal lattice polygons.

Enhanced understanding of lattice polygon properties.

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 $P\subset\mathbb{R}^2$ that involve the lattice size of $P$. In particular, we improve bounds established by Arnold, and B\'ar\'any and Pach. We also provide a classification of minimal lattice polygons $P\subset\mathbb{R}^2$ of fixed lattice size $\operatorname{ls_\square}(P)$.