1 − 2 + 3 − 4 + · · ·

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

$\underset{Y}{\overset{*}{\int }}{H}_{m-n}(A\cap f^{-1}(y))d{H}_n(y)\le \frac{v_{m-n}{v}_n}{v_m}(\text{Lip }f{)}^n{H}_m(A),$

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.

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.