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Bounds on the Geometric Complexity of Optimal Centroidal Voronoi Tesselations in 3D

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Abstract

Gersho’s conjecture in 3D asserts the asymptotic periodicity and structure of the optimal centroidal Voronoi tessellation. This relatively simple crystallization problem remains to date open. We prove bounds on the geometric complexity of optimal centroidal Voronoi tessellations as the number of generators tends to infinity. Combined with an approach of Gruber in 2D, these bounds reduce the resolution of the 3D Gersho’s conjecture to a finite, albeit very large, computation of an explicit convex problem in finitely many variables.

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Notes

  1. For example, Hales’ celebrated resolution of the Honeycomb Conjecture in [18].

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Acknowledgements

We thank the anonymous referees for their many comments which improved the manuscript; in particular, one of the referees provided us with the simple proof of (8) in Sect. 4.1. This work was begun while Lu was a postdoctoral fellow at McGill University. He would like to thank the CRM (Centre de Recherches Mathématique) for their partial support during this period, and Lakehead University for their partial support through its startup and RDF fundings. Both authors acknowledge the support of NSERC through their Discovery Grants Program. The authors would also like to thank David Bourne for his comments on a previous draft.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada

    Rustum Choksi & Xin Yang Lu

  2. Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON, Canada

    Xin Yang Lu

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  1. Rustum Choksi
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  2. Xin Yang Lu
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Correspondence to Rustum Choksi.

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Communicated by R. Seiringer

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Choksi, R., Lu, X.Y. Bounds on the Geometric Complexity of Optimal Centroidal Voronoi Tesselations in 3D. Commun. Math. Phys. 377, 2429–2450 (2020). https://doi.org/10.1007/s00220-020-03789-y

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