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Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert–Einstein Functional

Published online by Cambridge University Press:  20 November 2018

Ivan Izmestiev
Ivan Izmestiev*
Affiliation:
Institut für Mathematik, Freie Universität Berlin, Arnimallee 2, D-14195 Berlin, Germany. e-mail: izmestiev@math.fu-berlin.de
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Abstract

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The paper presents a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert–Einstein functional on the space of “warped polyhedra” with a fixed metric on the boundary.

The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas.

In the spherical space and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert–Einstein functional and the volume, as well as between both kinds of rigidity.

We review some of the related work and discuss directions for future research.

Keywords

52B9953C24convex polyhedronrigidityHilbert–Einstein functionalMinkowski theorem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 4 , 01 August 2014 , pp. 783 - 825
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

Supported by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 247029-SDModels.

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