On equidissection of balanced polygons

TL;DR

This paper proves that lattice balanced polygons with odd area cannot be divided into an odd number of equal-area triangles, extending previous results and exploring links to tropical geometry.

Contribution

It generalizes Monsky's theorem from squares to all balanced polygons and connects equidissection problems to tropical geometry.

Findings

Lattice balanced polygons of odd area cannot be dissected into an odd number of equal-area triangles.

Established links between equidissection problems and tropical geometry.

Extended Monsky's theorem to a broader class of polygons.

Abstract

In this paper we show that a lattice balanced polygon of odd area cannot be cut into an odd number of triangles of equal areas. First result of this type was obtained by Paul Monsky in 1970. He proved that a square cannot be cut into an odd number of triangles of equal areas. In 2000 Sherman Stein conjectured that the same holds for any balanced polygon. We also show connections between the equidissection problem and tropical geometry.