An isoperimetric inequality for planar triangulations

TL;DR

This paper establishes a discrete isoperimetric inequality for planar triangulations, demonstrating that hexagons in the triangular lattice maximize volume for a given boundary in minimal degree 6 triangulations.

Contribution

It introduces a discrete analogue of Weil's isoperimetric theorem, extending classical geometric inequalities to combinatorial triangulations.

Findings

Hexagons in the triangular lattice have maximal volume for fixed boundary.

The inequality applies to triangulations with minimal degree 6.

Provides a new discrete geometric inequality analogous to classical results.

Abstract

We prove a discrete analogue to a classical isoperimetric theorem of Weil for surfaces with non-positive curvature. It is shown that hexagons in the triangular lattice have maximal volume among all sets of a given boundary in any triangulation with minimal degree 6.