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Pseudo-Edge Unfoldings of Convex Polyhedra

  • Nicholas Barvinok
  • Mohammad GhomiEmail author
Ricky Pollack Memorial Issue
  • 5 Downloads

Abstract

A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable. The proof is based on a result of Pogorelov on convex caps with prescribed curvature, and an unfoldability obstruction for almost flat convex caps due to Tarasov. Our example, which has 340 vertices, significantly simplifies an earlier construction by Tarasov, and confirms that Dürer’s conjecture does not hold for pseudo-edge unfoldings.

Keywords

Edge unfolding Dürer conjecture Almost flat convex cap Prescribed curvature Weighted spanning forest Pseudo-edge graph Isometric embedding 

Mathematics Subject Classification

Primary: 52B10 53C45 Secondary: 57N35 05C10 

Notes

Acknowledgements

Our debt to the original investigations of A. Tarasov [24] is evident throughout this work. Thanks also to J. O’Rourke for his interest and useful comments on earlier drafts of this paper. Furthermore we are grateful to several anonymous reviewers who prompted us to clarify the exposition of this work. Parts of this work were completed while the first named author participated in the REU program in the School of Math at Georgia Tech in the Summer of 2017.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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