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Lifting degenerate simplices with a single volume constraint

  • Lizhao ZhangEmail author
Original Paper
  • 7 Downloads

Abstract

Let \(M^d\) be the spherical, Euclidean, or hyperbolic space of dimension \(d\ge n+1\). Given any degenerate \((n+1)\)-simplex \({\mathbf {A}}\) in \(M^d\) with non-degenerate n-faces \(F_i\), there is a natural partition of the set of n-faces into two subsets \(X_1\) and \(X_2\) such that \(\sum _{X_1}V_n(F_i)=\sum _{X_2}V_n(F_i)\), except for a special spherical case where \(X_2\) is the empty set and \(\sum _{X_1}V_n(F_i)=V_n({\mathbb {S}}^n)\) instead. For all cases, if the vertices vary smoothly in \(M^d\) with a single volume constraint that \(\sum _{X_1}V_n(F_i)-\sum _{X_2}V_n(F_i)\) is preserved as a constant (0 or \(V_n({\mathbb {S}}^n)\)), we prove that if a stress invariant \(c_{n-1}(\alpha ^{n-1})\) of the degenerate simplex is non-zero, then the vertices will be confined to a lower dimensional \(M^n\) for any sufficiently small motion. This answers a question of the author and we also show that in the Euclidean case, \(c_{n-1}(\alpha ^{n-1})=0\) is equivalent to the vertices of a dual degenerate \((n+1)\)-simplex lying on an \((n-1)\)-sphere in \({\mathbb {R}}^n\).

Keywords

Rigidity Lifting k-stress Dual Characteristic polynomial 

Mathematics Subject Classification

52C25 52B11 51F99 

Notes

Acknowledgements

One of the main results Theorem 1.4 was obtained during the author’s PhD thesis work at M.I.T.. I would like to thank Professor Kleitman and Professor Stanley for their helpful discussions. I would also like to thank Wei Luo and a referee for carefully reading the manuscript and making many suggestions.

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Copyright information

© The Managing Editors 2019

Authors and Affiliations

  1. 1.MadisonUSA

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