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Besicovitch inequality
 | This article may be too technical for most readers to understand. (April 2021) |
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 with a Riemannian metric . Let
![{\displaystyle [0,1]^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40160923273b7109968df994dca832b91d957bf2)
denote the distance between opposite faces of the cube. The Besicovitch inequality asserts that
![{\displaystyle \prod _{i}d_{i}\leq Vol([0,1]^{n},g)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd34910c6caecad3df9c481007232314618b8902)
The inequality can be generalized in the following way. Given an n-dimensional Riemannian manifold M with connected boundary and a smooth map , such that the restriction of f to the boundary of M is a degree 1 map onto , define
![{\displaystyle f:M\rightarrow [0,1]^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c1750d417f7a83802793ecee97072434f43806)
![{\displaystyle \partial [0,1]^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94db59861d7f3c3ebeaa4bc13794eb1093038903)
Then .
The Besicovitch inequality was used to prove systolic inequalities on surfaces.[2][3]
- ^ 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.
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