“Eilenberg's inequality”的意思、由来-开放百科全书

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Eilenberg's inequality

释义

References

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

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.

The Eilenberg's Inequality is a key ingredient for the proof of the Coarea formula.

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}}.

1 : Inequalities

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