Convex Lattice Polygons with $k\ge3$ Interior Points

TL;DR

This paper investigates the structure of convex lattice polygons with a fixed number of collinear interior points, providing a construction method, necessary conditions, and classifying possible interior point counts.

Contribution

It introduces a method to construct primitive lattice triangles from polygon edges and classifies integers for which such polygons exist.

Findings

A process to construct primitive lattice triangles from polygon edges.

Necessary conditions for constructing such primitive triangles.

Classification of integers for which convex polygons with given interior points exist.

Abstract

We study the geometry of convex lattice $n$-gons with $n$ boundary lattice points and $k\geq 3$ collinear interior lattice points. We describe a process to construct a primitive lattice triangle from an edge of a convex lattice $n$-gon, hence adding one edge in a way so that the number of boundary points increases by $1$, while the number of interior points remains unchanged. We also present the necessary conditions to construct such a primitive lattice triangle, as well as an upper bound for the number of times this is possible. Finally, we apply the previous results to fully classify the positive integers for which there exists a convex $n$-gon with $k$ collinear and non-collinear interior points.