Besicovitch inequality

In mathematics, the Besicovitch inequality is a geometric inequality relating volume of a set and distances between certain subsets of its boundary. The inequality was first formulated by Abram Besicovitch.^[1]

Consider the n-dimensional cube $[0,1]^{n}$ with a Riemannian metric $g$ . Let

d_{i}=dist_{g}(\{x_{i}=0\},\{x_{i}=1\})

denote the distance between opposite faces of the cube. The Besicovitch inequality asserts that

\prod _{i}d_{i}\leq Vol([0,1]^{n},g)

The inequality can be generalized in the following way. Given an n-dimensional Riemannian manifold M with connected boundary and a smooth map $f:M\rightarrow [0,1]^{n}$ , such that the restriction of f to the boundary of M is a degree 1 map onto $\partial [0,1]^{n}$ , define

d_{i}=dist_{M}(f^{-1}(\{x_{i}=0\}),f^{-1}(\{x_{i}=1\}))

Then $\prod _{i}d_{i}\leq Vol(M)$ .

The Besicovitch inequality was used to prove systolic inequalities on surfaces.^[2]^[3]

Notes

^ A. S. Besicovitch, On two problems of Loewner, J. London Math. Soc. 27 (1952) 141–144.
^ Mikhael Gromov. Filling Riemannian manifolds. J. Differential Geom. 18 (1983), no. 1, 1-147. doi:10.4310/jdg/1214509283
^ P. Papasoglu, Cheeger constants of surfaces and isoperimetric inequalities, Trans. Amer. Math. Soc. 361 (2009) 5139–5162.

References

Burago, Dmitri & Burago, Yuri & Ivanov, Sergei. (2001). A Course in Metric Geometry. Graduate Studies in Mathematics 33.
Burago Yu. & Zalgaller, V. A. Geometric inequalities. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 285. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1988.
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. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. xx+585 pp. ISBN 0-8176-3898-9.
Burago, D., & Ivanov, S. (2002). On Asymptotic Volume of Finsler Tori, Minimal Surfaces in Normed Spaces, and Symplectic Filling Volume. Annals of Mathematics, 156(3), second series, 891-914. doi:10.2307/3597285

[1] A. S. Besicovitch, On two problems of Loewner, J. London Math. Soc. 27 (1952) 141–144.

[2] Mikhael Gromov. Filling Riemannian manifolds. J. Differential Geom. 18 (1983), no. 1, 1-147. doi:10.4310/jdg/1214509283

[3] P. Papasoglu, Cheeger constants of surfaces and isoperimetric inequalities, Trans. Amer. Math. Soc. 361 (2009) 5139–5162.

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