Hand lemma

The arm lemma is a lemma in the proof of the Cauchy polyhedron theorem .

Informally, a statement can be described as follows: Imagine a robot arm consisting of several links connected by joints. Each link is a segment, and the whole arm is broken . Let the entire arm of the robot can move in one plane. Suppose, in the initial state, the robot arm forms a convex broken line, that is, such a broken line, that if we connect the ends of the broken line, we get a convex polygon . Suppose now that the robot increases the angle in each joint. The lemma states that then the distance between the beginning and the end of the arm will increase.

Despite the simplicity of the statement, the proof of the lemma is not simple. In particular, in this place the original proof of Cauchy has an error. This mistake went unnoticed for more than a hundred years. It was noticed by Ernst Steinitz , apparently, between 1920 and 1928 and corrected only in 1934 ^[1] .

Content

Wording

Suppose $A_{1}A_{2}\dots A_{n}$ ${\ displaystyle A_ {1} A_ {2} \ dots A_ {n}}$ convex polygon on the Euclidean plane and $B_{1}B_{2}\dots B_{n}$ ${\ displaystyle B_ {1} B_ {2} \ dots B_ {n}}$ a broken line in a plane or space such that

$A_{i}A_{i+1}=B_{i}B_{i+1}$ ${\ displaystyle A_ {i} A_ {i + 1} = B_ {i} B_ {i + 1}}$ at $i<n$ ${\ displaystyle i <n}$ ,
$\measuredangle A_{i-1}A_{i}A_{i+1}\leq \measuredangle B_{i-1}B_{i}B_{i+1}$ ${\ displaystyle \ measuredangle A_ {i-1} A_ {i} A_ {i + 1} \ leq \ measuredangle B_ {i-1} B_ {i} B_ {i + 1}}$ at $1<i<n$ ${\ displaystyle 1 <i <n}$ .

Then

A_{1}A_{n}\leq B_{1}B_{n}.

{\ displaystyle A_ {1} A_ {n} \ leq B_ {1} B_ {n}.}

Moreover, in case of equality, broken lines $A_{1}A_{2}\dots A_{n}$ ${\ displaystyle A_ {1} A_ {2} \ dots A_ {n}}$ and $B_{1}B_{2}\dots B_{n}$ ${\ displaystyle B_ {1} B_ {2} \ dots B_ {n}}$ congruent.

Variations and generalizations

A similar result is true on the sphere and the Lobachevsky plane .
Zalgaller 's theorem . If two spherical $n$ ${\ displaystyle n}$ -Golnikov $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ respective sides are equal and polygon $A$ ${\ displaystyle A}$ lies in the hemisphere, then at least one of the corners $B$ ${\ displaystyle B}$ not less than the corresponding angle $A$ ${\ displaystyle A}$ . ^[2]

Bent bow lemma ^[3] - version of the arm lemma for smooth curves: Let $\gamma$ ${\ displaystyle \ gamma}$ and ${\tilde {\gamma }}$ ${\ displaystyle {\ tilde {\ gamma}}}$ - a pair of smooth curves parametrized by the length defined on the same interval $[a,b]$ ${\ displaystyle [a, b]}$ . Suppose for any $t$ ${\ displaystyle t}$ inequality holds $\kappa _{\gamma }(t)\leq \kappa _{\tilde {\gamma }}(t)$ ${\ displaystyle \ kappa _ {\ gamma} (t) \ leq \ kappa _ {\ tilde {\ gamma}} (t)}$ where $\kappa _{\gamma }(t)$ ${\ displaystyle \ kappa _ {\ gamma} (t)}$ and $\kappa _{\tilde {\gamma }}(t)$ ${\ displaystyle \ kappa _ {\ tilde {\ gamma}} (t)}$ denotes curvature $\gamma$ ${\ displaystyle \ gamma}$ and correspondingly ${\tilde {\gamma }}$ ${\ displaystyle {\ tilde {\ gamma}}}$ at $t$ ${\ displaystyle t}$ . Further suppose that ${\tilde {\gamma }}$ ${\ displaystyle {\ tilde {\ gamma}}}$ there is an arc of a planar convex curve, that is, it passes along the boundary of some convex planar figure. Then the distance between the ends $\gamma$ ${\ displaystyle \ gamma}$ does not exceed the distance between the ends ${\tilde {\gamma }}$ ${\ displaystyle {\ tilde {\ gamma}}}$ ; i.e,
$|\gamma (b)-\gamma (a)|\geq |{\tilde {\gamma }}(b)-{\tilde {\gamma }}(a)|.$ ${\ displaystyle | \ gamma (b) - \ gamma (a) | \ geq | {\ tilde {\ gamma}} (b) - {\ tilde {\ gamma}} (a) |.}$

(The lemma is true if

\gamma

{\ displaystyle \ gamma}

there is a curve in Euclidean space of arbitrary dimension.)

Notes

↑ Steinitz E., Rademacher H. Vorlesungen ̈uber die Theorie der Polyeder. Berlin: Springer-Verl., 1934.
↑ V.A. Zalgaller . On deformations of a polygon on a sphere // Uspekhi Mat . - 1956. - T. 11 , No. 5 (71) . - S. 177-178 .
↑ Toponogov, V. A. Differential geometry of curves and surfaces . - Fizmatkniga, 2012. - ISBN 978-5-89155-213-5 .

Literature

I. Kh. Sabitov , Around the proof of the Legendre - Cauchy lemma on convex polygons Sibirsk. mate. Zh., 2004, Volume 45, No. 4, p. 892-919
Lecture 24 in Tabachnikov S.L., Fuchs D.B. Mathematical divertissement. - ICMMO, 2011 .-- ISBN 978-5-94057-731-7 .

[1] Steinitz E., Rademacher H. Vorlesungen ̈uber die Theorie der Polyeder. Berlin: Springer-Verl., 1934.

[2] V.A. Zalgaller . On deformations of a polygon on a sphere // Uspekhi Mat . - 1956. - T. 11 , No. 5 (71) . - S. 177-178 .

[3] Toponogov, V. A. Differential geometry of curves and surfaces . - Fizmatkniga, 2012. - ISBN 978-5-89155-213-5 .