Eilenberg's inequality

Eilenberg's inequality is a mathematical inequality for Lipschitz-continuous functions.

Let ƒ : X → Y be a Lipschitz-continuous function between metric spaces whose Lipschitz constant is denoted by Lip ƒ. Then, Eilenberg's inequality states that

${\displaystyle \int _{Y}^{*}H_{m-n}(A\cap f^{-1}(y))\,dH_{n}(y)\leq {\frac {v_{m-n}v_{n}}{v_{m}}}({\text{Lip }}f)^{n}H_{m}(A),}$

for any A ⊂ X and all 0 ≤ n ≤ m, where

• the asterisk denotes the upper Lebesgue integral,
• v_n is the volume of the unit ball in Rⁿ,
• H_n is the n-dimensional Hausdorff measure.

The Eilenberg's Inequality is a key ingredient for the proof of the Coarea formula. Indeed, it confirms the coarea formula when A is a set of measure zero, which allows one to ignore from the domain any annoying null set, for example the set where a Lipschitz function is not differentiable.

In many texts it is stated with some restriction on the metric spaces, but this is unnecessary. A full proof without any conditions on the metric spaces can be found in Reichel's PhD thesis referenced below. A new proof of the general case can be found in 2020 paper Esmayli, Behnam & Hajłasz, Piotr. (2020). The Coarea Inequality (Arxiv Link).

References

• Yu. D. Burago and V. A. Zalgaller, Geometric inequalities. Translated from the Russian by A. B. Sosinskiĭ. Springer-Verlag, Berlin, 1988. ISBN 3-540-13615-0.

• (Doctoral Thesis) Reichel, Lorenz Philip, The coarea formula for metric space valued maps, 2009 https://doi.org/10.3929/ethz-a-005905811