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
for any A ⊂ X and all 0 ≤ n ≤ m, where
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. This allows the proof to ignore null sets as is a necessary step in many proofs in analysis.
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.