Convex Lattice Polygons with $k\ge3$ Interior Points

TL;DR

The paper investigates convex lattice polygons with $B=n$ and at least $k\ge3$ collinear interior points, introducing a constructive method to append primitive lattice triangles to edges that preserves the interior point count while increasing the boundary by one. It formalizes an equivalence relation via integral unimodular affine transformations and uses isometries to handle edge rotations, establishing conditions under which such appends maintain convexity and providing an upper bound on the number of allowable augmentations. The core results classify when convex $n$-gons with collinear interior points exist, proving that only $n\in\{3,4,5,6\}$ can occur, and show that for $k\ge4$ there exist polygons with non-collinear interior points for these same $n$, with explicit constructions for $4$-gons. The work combines Bézout-type arguments, Pick's theorem, and half-plane analyses to bound configurations and to relate collinear interior points to canonical representations on the $x$-axis, yielding a complete existence/absence landscape for the problem.

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.

Figures (2)

• Figure 1: The standard orientation of an edge $L_i$.
• Figure 2: Geometric interpretation of appending primitive triangles to the upper and lower edges of the previously appended primitive triangle.

Theorems & Definitions (20)

• Theorem
• Theorem
• Lemma 1
• proof
• Theorem 2: martin
• Lemma 3
• proof
• Lemma 4
• proof
• Example 1