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The deformation space of Delaunay triangulations of the sphere

Yanwen Luo, Tianqi Wu and Xiaoping Zhu

Vol. 323 (2023), No. 1, 115–127
DOI: 10.2140/pjm.2023.323.115
Abstract

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.

Keywords
geodesic triangulations, angle structures, Delaunay triangulations, spaces of embeddings
Mathematical Subject Classification
Primary: 05C62, 58D10
Milestones
Received: 22 April 2022
Revised: 21 November 2022
Accepted: 7 March 2023
Published: 29 May 2023
Authors
Yanwen Luo
Department of Mathematics
Rutgers University
Piscataway, NJ
United States
Tianqi Wu
Department of Mathematics
Clark University
Worcester, MA
United States
Xiaoping Zhu
Department of Mathematics
Rutgers University
New Brunswick, NJ
United States

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