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 $$[0,1]^n$$ with a Riemannian metric $$g$$. Let $d_i= dist_g(\{x_i=0\}, \{x_i=1\})$ denote the distance between opposite faces of the cube. The Besicovitch inequality asserts that $\prod_i d_i \geq Vol([0,1]^n,g)$ The inequality can be generalized in the following way. Given an n-dimensional Riemannian manifold M with connected boundary and a smooth map $$f: M \rightarrow [0,1]^n$$, such that the restriction of f to the boundary of M is a degree 1 map onto $$ \partial [0,1]^n$$, define $d_i= dist_M(f^{-1}(\{x_i=0\}), f^{-1}(\{x_i=1\}))$ Then $$\prod_i d_i \geq Vol(M)$$.

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