Eilenberg's inequality - Alchetron, the free social encyclopedia

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

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

∫ Y ∗ H m − n ( A ∩ f − 1 ( y ) ) d H n ( y ) ≤ v m − n v n v m ( 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.

References

Eilenberg's inequality Wikipedia

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