# Applications of Cauchy's Arm Lemma

Cauchy's Arm Lemma is used in the proof of Cauchy's Rigidity Theorem for convex polyhedra. The Lemma states that in the plane or on the sphere that if all but one of the side lengths of two convex polygons $P$ and $P'$ are the same, and the angles formed by the remaining sides of P are less than or equal to those of the remaining sides of $P'$, then the ommitted side length from $P$ is less than or equal to the omitted side length for $P'$, with equality occurring iff the angles are all the same. I know of a couple of extensions of this theorem (a nice presentation of this sort of thing is available in O'Rourke's paper [O'R01]). The Lemma can also be used to show that convex linkages may be (unsurprisingly) straightened.

I'm seeking other applications of this Lemma, particularly those that might be suitable for use as exercises in an advanced undergraduate course, or other applications in the theory of polyhedra.

[O’R01] Joseph O’Rourke, An extension of Cauchy’s arm lemma with application to curve development, Discrete and computational geometry (Tokyo, 2000), Lecture Notes in Comput. Sci., vol. 2098, Springer, Berlin, 2001, pp. 280–291. MR MR2043660 (2004m:52052)

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## 2 Answers

The question is a little too general to have a single answer. Cauchy's arm lemma is a basic technical result in rigidity theory. One reason it has a name is because it is intuitively obvious, but the most natural proof Cauchy originally came up with is false.

Now, other than Cauchy theorem, it has few direct consequences. A.D. Alexandrov developed a whole family of results similar in nature (say, with angles preserved but polygon lengths extended), which proved new results on rigidity. This was, e.g. his way of extending the Minkowski uniqueness theorem in $\Bbb R^3$ to polytopes with (say) equal normals and perimeters of faces. You can read it in Alexandrov's famous monograph. Sabitov made a historical study behind the original Cauchy lemma and proved a number of related results as well.

Let me emphasize that from a modern point of view, the Arm Lemma is a "relative" version of the discrete four vertex theorem, which is a fundamental result in its own right. I explain the background, connections and variations in my book, sections 21-23. Note that this is not the only "relative result" of this type: in addition to Alexandrov's lemmas, see Tabachnikov's theorem in 21.6. See also numerous historical and other references mentioned there. In conclusion, much of this is relatively easy but not straigtforward. I think at least some of these results fit well to an undergrad course.

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If I may add one explicit example as a late-comer, even though it is along the lines with which you are already familiar: Cauchy's arm lemma may be used to prove that the curve that is the intersection of a plane with a convex polyhedron develops on a plane without self-intersection. This is described in Geometric Folding Algorithms: Linkages, Origami, Polyhedra, p.377ff.

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