Waist of the sphere theorem

In mathematics , the so-called waist of the sphere theorem (German about "theorem about the waist of the sphere"), also called waist inequality , is an inequality of Euclidean geometry. It was proven by the Russian-French mathematician Michail Leonidowitsch Gromow .

Inequality

Be the unit sphere and ${\displaystyle S^{n}\subset \mathbb {R} ^{n+1}}$

${\displaystyle f\colon S^{n}\rightarrow \mathbb {R} ^{q}}$

a continuous map, . ${\displaystyle q\leq n}$

Then there is at least one with ${\displaystyle y\in \mathbb {R} ^{q}}$

${\displaystyle \mathrm {vol} (U_{\varepsilon }(f^{-1}(y)))\geq \mathrm {vol} (U_{\varepsilon }(S^{n-q}))\ {\text{für alle}}\ \varepsilon >0}$.

Here the environment and the dimensional equator . ${\displaystyle U_{\varepsilon }}$ ${\displaystyle \varepsilon }$ ${\displaystyle S^{n-q}\subset S^{n}}$ ${\displaystyle (n-q)}$

Combinatorial version

For every continuous mapping of a simplex into the there is a with ${\displaystyle F\colon \Delta ^{N-1}\to \mathbb {R} ^{q}}$ ${\displaystyle (N-1)}$ ${\displaystyle \mathbb {R} ^{q}}$ ${\displaystyle y\in \mathbb {R} ^{q}}$

${\displaystyle \mid F^{-1}(y)\mid \geq c(q){\dbinom {N}{q+1}}}$

for a constant that only depends on . ${\displaystyle q}$ ${\displaystyle c(q)}$

In particular, there are ever points in a point , in at least the of these spanned points is -Simplizes. (This is a generalization of Carathéodory's theorem, which goes back to Bárány .) ${\displaystyle N}$ ${\displaystyle \mathbb {R} ^{q}}$ ${\displaystyle y\in \mathbb {R} ^{q}}$ ${\displaystyle c(q){\tbinom {N}{q+1}}}$ ${\displaystyle {\tbinom {N}{q+1}}}$ ${\displaystyle N}$ ${\displaystyle q}$

history

The inequality of the mapping under certain regularity requirements can be proven with the methods of geometric measurement theory developed by Almgren in 1965 , but was not mentioned in this form by Almgren himself. In 1983 Gromov first gave a brief geometric proof for the existence of a non-explicit lower bound of and finally in 2003 a proof of the inequality in the above form using algebraic topology. Detailed evidence was published by Memarian in 2011. ${\displaystyle f}$ ${\displaystyle \sup _{y\in \mathbb {R} ^{q}}{\mathcal {H}}^{n-q}(f^{-1}(y))}$

literature

• Misha Gromov : Metric structures for Riemannian and non-Riemannian spaces . Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Reprint of the 2001 English edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2007 (Section 2.12 ½)

Web links

• Larry Guth : The waist inequality in Gromov's work pdf (appears in: The Abel Prize 2008-2012 , Springer Verlag 2014, ISBN 978-3-642-39448-5 )
• Parker Glynn-Addey: Of waists and spheres
• Arseniy Akopyan, Alfredo Hubard, Roman Karasev: Lower and upper bounds for the waists of different spaces pdf

Individual evidence

1. ↑ FJAlmgren: The theory of Varifolds - a variational calculus in the large for the k-dimensional area integrand . Mimeographed notes, 1965.
2. ↑ Mikhael Gromov: Filling Riemannian manifolds. J. Differential Geom. 18 (1983) no. 1, 1-147. (Appendix 1 (F)) pdf
3. ↑ M.Gromov: Isoperimetry of waists and concentration of maps. Geom. Funct. Anal. 13 (2003), no. 1, 178-215. pdf ( Memento of the original from September 24, 2015 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
4. ↑ Yashar Memarian: On Gromov's waist of the sphere theorem. J. Topol. Anal. 3 (2011), no. 1, 7-36. pdf