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 ${\displaystyle [0,1]^{n}}$ 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.[2][3]