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'''Eilenberg's inequality''' is a [[inequality (mathematics)|mathematical inequality]] for [[Lipschitz continuity|Lipschitz-continuous function]]s.

Let ''&fnof;''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''Y'' be a Lipschitz-continuous function between [[separable space|separable]] [[metric space]]s whose Lipschitz constant is denoted by Lip&nbsp;''&fnof;''. Then, Eilenberg's inequality states that

:<math>\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), </math>

for any ''A''&nbsp;⊂&nbsp;''X'' and all 0&nbsp;≤&nbsp;''n''&nbsp;≤&nbsp;''m'', where

* the asterisk denotes the upper&nbsp;[[Lebesgue integral]],
* ''v''<sub>''n''</sub> is the volume of the unit ball in&nbsp;'''R'''<sup>''n''</sup>,
* ''H''<sub>''n''</sub> 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.

[[Category:Inequalities]]