The Deformation Space of Delaunay Triangulations of the Sphere

TL;DR

This paper investigates the topology of spaces of Delaunay triangulations on the sphere, confirming they share the same homotopy type as smooth diffeomorphism groups, unlike in flat tori or polygons.

Contribution

It proves the homotopy equivalence of Delaunay triangulation spaces and diffeomorphism groups for the sphere, extending understanding of their topological structure.

Findings

Spaces of Delaunay triangulations on the sphere have the same homotopy type as diffeomorphism groups.

The conjecture does not hold for flat tori and convex polygons, which can have disconnected triangulation spaces.

Abstract

In this paper, we determine the topology of the spaces of convex polyhedra inscribed in the unit $2$-sphere and the spaces of strictly Delaunay geodesic triangulations of the unit $2$-sphere. These spaces can be regarded as discretized groups of diffeomorphisms of the unit $2$-sphere. Hence, it is natural to conjecture that these spaces have the same homotopy types as those of their smooth counterparts. The main result of this paper confirms this conjecture for the unit $2$-sphere. It follows from an observation on the variational principles on triangulated surfaces developed by I. Rivin. On the contrary, the similar conjecture does not hold in the cases of flat tori and convex polygons. We will construct simple examples of flat tori and convex polygons such that the corresponding spaces of Delaunay geodesic triangulations are not connected.