Bounds on Area Involving Lattice Size

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 Bárány and Pach. We also provide a classification of minimal lattice polygons $P\subset {\mathbb{R}}^2$ of fixed lattice size $\mathrm{l}{\mathrm{s}}_{\square }(P)$.

15 pages, 12 figures

References in corpus (3)

• Small polygons and toric codes
• Lattice Size of Width One Lattice Polytopes in ${\mathbb{R}}^3$
• Lattice Size and Generalized Basis Reduction in Dimension 3

Cited by in corpus (1)

• Lattice Size in Higher Dimension