# 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.

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

$\displaystyle{ d_i= dist_g(\{x_i=0\}, \{x_i=1\}) }$

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

$\displaystyle{ \prod_i d_i \geq Vol([0,1]^n,g) }$

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

$\displaystyle{ d_i= dist_M(f^{-1}(\{x_i=0\}), f^{-1}(\{x_i=1\})) }$

Then $\displaystyle{ \prod_i d_i \geq Vol(M) }$.

The Besicovitch inequality was used to prove systolic inequalities on surfaces.