Eilenberg's inequality

Eilenberg's inequality

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}(Acap f^{-1}(y)) , dH_n(y) leq frac{v_{m-n}v_n}{v_m}( ext{Lip }f)^n H_m(A), for any "A" ⊂ "X" and all 0 ≤ "n" ≤ "m". Here the asterisk denotes the upper Lebesgue integral, and "v""n" is the volume of the unit ball in R"n".

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.


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