Eilenberg's inequality

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

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

\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. Here the asterisk denotes the upper Lebesgue integral, and vn is the volume of the unit ball in Rn.

[edit] 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.