Bounding the area of a centered dual two-cell below, given lower bounds on its side lengths

TL;DR

This paper provides a computable lower bound on the area of a hyperbolic two-cell in the centered dual decomposition, based on lower bounds of its edge lengths, with an implementation for small cases.

Contribution

It introduces a method to effectively compute area bounds of hyperbolic two-cells given edge length constraints, extending prior work on the centered dual decomposition.

Findings

Provides an explicit formula for area bounds

Includes Python code for computation when n<10

Bounds are sharp or near-sharp for reasonable edge lengths

Abstract

Suppose $C$ is a compact, $n$-edged two-cell of the centered dual decomposition of a locally finite set in the hyperbolic plane, a coarsening of the Delaunay tessellation which was introduced in the author's prior work. We describe an effectively computable lower bound on the area of $C$, given an $n$-tuple of positive real numbers bounding the lengths of the edges of $C$ below. The ancillary materials contain Python code implementing (for $n<10$) an algorithm to compute this bound. For geometrically reasonable edge length bounds, we expect the given area bound to be sharp or near-sharp.