Given a Lipschitz map f from a cube into a metric space, we find several equivalent conditions for f to have a Lipschitz factorization through a metric tree. As an application we prove a recent conjecture of David and Schul. The techniques developed for the proof of the factorization result yield several other new and seemingly unrelated results. We prove that if f is a Lipschitz mapping from an open set in R n onto a metric space X, then the topological dimension of X equals n if and only if X has positive n-dimensional Hausdorff measure. We also prove an area formula for length-preserving maps between metric spaces, which gives, in particular, a new formula for integration on countably rectifiable sets in the Heisenberg group.

The aim of this paper is to provide a self-contained proof of a general case of the coarea inequality, also known as the Eilenberg inequality. The result is known, but we are not aware of any place that a proof would be written with all details. The known proof is based on a difficult result of Davies. Our proof is elementary and does not use Davies' theorem. Instead we use an elegant argument that we learned from Nazarov through MathOverflow. We also obtain some generalizations of the coarea inequality.Here H α stands for the α-dimensional Hausdorff measure and´ * g dµ is the upper integral which does not require measurability of the integrand.Remark 1.2. In general, we cannot expect measurability of the function (1.2) as the following simple example shows: Let V ⊂ R be a non-measurable set. Let X = V , Y = R and f : X → Y , f (x) = x. Then for s = t = 1, and E = X, the function (1.2) is the characteristic function of V and therefore is not measurable. It was communicated to us by Mattila [23] that (1.2) is measurable with respect to the sigma-algebra generated by analytic sets if X and Y are Polish spaces and E is analytic. This is a consequence of the work of Dellacherie [6], see Remark 7.8 in [24]. However, we did not verify this statement.

Given a Lipschitz map 𝑓 from a cube into a metric space, we find several equivalent conditions for 𝑓 to have a Lipschitz factorization through a metric tree. As an application, we prove a recent conjecture of David and Schul. The techniques developed for the proof of the factorization result yield several other new and seemingly unrelated results. We prove that if 𝑓 is a Lipschitz mapping from an open set in ℝ 𝑛 onto a metric space 𝑋, then the topological dimension of 𝑋 equals 𝑛 if and only if 𝑋 has positive 𝑛-dimensional Hausdorff measure. We also prove an area formula for length-preserving maps between metric spaces, which gives, in particular, a new formula for integration on countably rectifiable sets in the Heisenberg group.

We study coarea inequalities for metric surfaces — metric spaces that are topological surfaces, without boundary, and which have locally finite Hausdorff 2-measure H 2 \mathcal {H}^2 . For monotone Sobolev functions u : X → R u\colon X \to \mathbb {R} , we prove the inequality ∫ R ∗ ∫ u − 1 ( t ) g d H 1 d t ≤ κ ∫ X g ρ d H 2 for every Borel  g : X → [ 0 , ∞ ] , \begin{equation*} \int _{ \mathbb {R} }^{*} \int _{ u^{-1}(t) } g \,d\mathcal {H}^{1} \,dt \leq \kappa \int _{ X } g \rho \,d\mathcal {H}^{2} \quad \text {for every Borel $g \colon X \rightarrow \left [0,\infty \right ]$,} \end{equation*} where ρ \rho is any integrable upper gradient of u u . If ρ \rho is locally L 2 L^2 -integrable, we obtain the sharp constant κ = 4 / π \kappa =4/\pi . The monotonicity condition cannot be removed as we give an example of a metric surface X X and a Lipschitz function u : X → R u \colon X \to \mathbb {R} for which the coarea inequality above fails.