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