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