# Coarea inequality, Eilenberg inequality

The general statement of the coarea inequality known also as the Eilenberg inequality is:

Theorem. If $$f:X\to Y$$ is a Lipschitz map between metric spaces and $$A\subset X$$, $$0\leq m\leq n$$, then $$\int_Y^*\mathcal{H}^{n-m}(f^{-1}(y)\cap A)\, d\mathcal{H}^m(y)\leq (\operatorname{Lip} f)^m\frac{\omega_{n-m}\omega_m}{\omega_n}\mathcal{H}^n(A).$$

Federer proved it (Theorem 2.10.25 in [2]) under additional assumptions. Then Davies [1] showed that these additional assumptions are not necessary and a reader may find a detailed proof in [3].

For a related discussion, see Open problems in Federer's Geometric Measure Theory

My questions is that I am not exactly sure what are the assumptions in the theorem:

Question 1. Is it assumed here that the spaces $$X$$ and $$Y$$ are separable?

Question 2. Are $$0\leq m\leq n$$ any real numbers of just integers?

[1] R. O. Davies, Increasing sequences of sets and Hausdorff measure, Proc. London Math. Soc. 20(1970), 222-236.

[2] H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften,Band 153 Springer-Verlag New York Inc., New York 1969.

[3] L. P. Reichel, The coarea formula for metric space valued maps, Ph.D. thesis, ETH Z ̈urich, 2009. http://e-collection.library.ethz.ch/eserv/eth:289/eth-289-02.pdf.

Edit. $$X$$ and $$Y$$ can be any metric spaces and $$0\leq m\leq n$$ any real numbers. For a complete and detailed proof as well as historical account see:

[4] B. Esmayli, P. Hajlasz, The Coarea Inequality. https://arxiv.org/abs/2006.00419.

The proof presented in [4] is elementary thanks to Fedja's help through Mathoverflow: Bounding an "integral" from below by the Hausdorff measure of the domain.

• I think it is true in the generality stated not assuming separability or integer exponents. For Question 1, if it is true for separable spaces, then the general case follows because if A is not separable, then the right-hand-side of the inequality is infinite anyway. Neither in Federer or the thesis of Reichel I could find any use of integer exponents. (In the thesis of Reichel, integer exponents are mentioned once following Definition 7.3, but this can be defined for non-integers as well). But I can't say it for sure... – rozu Mar 5 '19 at 14:31
• @rozu I also thought that Reichel never used integer values of $m$ and $n$, but if I correctly remember he defined $\omega_n$ only for integer value of $n$. That puzzled me. I am quite surprised that this so important and beautiful inequality has never been published with all details. – Piotr Hajlasz Mar 5 '19 at 18:10
• The values $\omega_k$ in the inequality should just be the multipliers to the diameter/2-expression that one puts in the definition of the corresponding Hausdorff measure, whatever one prefers for nonintegers. I agree completely, this should be published with details. I saw the coarea inequality used in literature also for noninteger exponents. But I don't remember where. – rozu Mar 6 '19 at 9:55
• @rozu "I agree completely, this should be published with details." Now it is. – Piotr Hajlasz Oct 13 '20 at 19:56

The theorem, as stated, is true for arbitrary metric spaces and for any pair of non-negative real numbers. Precisely,

Theorem (Co-area Inequality). If $$f:X\to Y$$ is a Lipschitz map between any metric spaces and an $$0\leq m\leq n < \infty$$ are any any pair of real numbers, then for any $$A \subset X$$, $$\int_Y^*\mathcal{H}^{n-m}(f^{-1}(y)\cap A)\, d\mathcal{H}^m(y)\leq (\operatorname{Lip} f)^m\frac{\omega_{n-m}\omega_m}{\omega_n}\mathcal{H}^n(A) \, .$$

Federer [1, 2.10.25] proved the theorem under the assumption that $$Y$$ is boundedly compact, that is, every bounded and closed set in $$Y$$ is compact. He needed this assumption because he proved the following lemma [1, 2.10.22] only under the same restriction. The lemma is needed in the proof of the coarea inequality.

Lemma (increasing sets lemma). Let $$(X,d)$$ be any metric space, $$0 \leq s < \infty$$ and $$0 < \delta < \infty$$. Then for any increasing set of subset $$E_1 \subset E_2 \subset \cdots$$, $$\mathcal{H}^s_\delta (\bigcup_{i=1}^\infty E_i) = \lim_{i \to \infty} \mathcal{H}^s_\delta (E_i) \, .$$

Years later, Davies proved this lemma under no restrictions on the metric space and commented that Federer tells him that this makes the extra assumptions for his co-area inequality "superfluous."

A well-written exposition of the proof of the co-area inequality can be found in Reichel's thesis [2], where he follows Federer's original proof while using the result of Davies.

[1] H. Federer, Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften,Band 153 Springer-Verlag New York Inc., New York 1969.

[2] L. P. Reichel, The coarea formula for metric space valued maps, Ph.D. thesis, ETH Z ̈urich, 2009. http://e-collection.library.ethz.ch/eserv/eth:289/eth-289-02.pdf