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]