Jarnik's convex lattice $n$-gon for non-symmetric norms

TL;DR

This paper extends Jarnik's classical problem by determining the minimum perimeter of convex lattice n-gons under general, possibly non-symmetric norms, broadening the understanding of lattice geometry.

Contribution

It provides a solution to the minimum perimeter problem for convex lattice polygons under non-symmetric norms, generalizing Jarnik's original symmetric norm case.

Findings

Established the minimum perimeter for convex lattice n-gons under non-symmetric norms.

Extended classical results to a broader class of norms.

Provided explicit constructions or bounds for minimal polygons.

Abstract

What is the minimum perimeter of a convex lattice $n$-gon? This question was answered by Jarnik in 1926. We solve the same question in the case when perimeter is measured by a (not necessarily symmetric) norm.