# Alexandrov's generalization of Cauchy's rigidity theorem

Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions.

The relevant statement in the article is not linked to any source. The sources at the end of the Wikipedia page seem to be only about $$3$$-dimensional polyhedra as well, in particular Alexandrov's book "Convex polyhedra".

Where can I find a reference for that statement?

The following is Theorem 27.2 of Igor Pak's book Lectures on Discrete and Polyhedral Geometry (which in general is a very nice resource for these sorts of questions):

Let $$P,Q\subset\mathbb{R}^d$$ (or $$P,Q \subset S^d_+$$), $$d\geq3$$ be two combinatorially equivalent (spherical) convex polyhedra whose corresponding facets are isometric. Then $$P$$ and $$Q$$ are isometric.

(Here $$S^d_+$$ is a d-dimensional hemisphere.)

• And Corollary 27.3 (p.256) is: Every convex polytope $d \ge 3$ is rigid. – Joseph O'Rourke May 20 '19 at 13:03

This may help:

Bauer, C. "Infinitesimal Rigidity of Convex Polytopes." Discrete Comput Geom (1999) 22: 177. https://doi.org/10.1007/PL00009453

"Aleksandrov [1] proved that a simple convex $$d$$-dimensional polytope, $$d \ge 3$$, is infinitesimally rigid if the volumes of its facets satisfy a certain assumption of stationarity. We extend this result..."

[1] is the 1958 Convex Polyhedra book.

• Thanks. Just to be sure, does infinitesimal rigidity imply global rigidity? – M. Winter May 20 '19 at 12:12
• @M.Winter: Not always, but I believe in this case Yes. – Joseph O'Rourke May 20 '19 at 13:04
• However, this is not a generalization of Cauchy rigidity theorem, but rather an infinitesimal version of a theorem by Minkowski on existence and uniqueness of polytopes with given facet volumes and normals. Minkowski proved global rigidity. – Ivan Izmestiev May 20 '19 at 15:39

Wikipedia is correct. This is discussed in Alexandrov's book "Convex polyhedra" in Section 3.6.5.

• Thank you a lot for the hint. I looked it up. I would probably not have recognized this short paragraph as a proof of the generalized theorem. Anyway, Wikipedia should link this statement to this source more directly. – M. Winter May 20 '19 at 21:08
• @M.Winter Wikipedia is, as the name suggests, a wiki ;) - so it's everyones job to make sure mistakes or oversights are corrected. Why not add the source reference yourself ? – Torque May 21 '19 at 8:37
• @Torque Actually, I thought about it. ;) But as everything that is public and not completely obvious how to do right, I hesitated. I will see what I can do! – M. Winter May 21 '19 at 8:50