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Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.
3
votes
0
answers
195
views
Hausdorff measure of the unit ball of a norm on $\mathbb{R}^n$ is a universal constant
In [1], Kirchheim proved the area formula for Lipschitz maps $f\colon \mathbb{R}^n\to X$ where $X$ is an arbitrary metric space, using the notion of metric differentiability. The metric derivative of …
4
votes
0
answers
769
views
Equality of Hausdorff measure and Lebesgue measure on manifolds (reference)
Let $\mathcal{M} \subset \mathbb{R}^N$ be an $n$-dimensional $C^1$ submanifold (connected). We have two metric functions on $\mathcal{M}$:
The Euclidean distance inherited from $\mathbb{R}^N$.
The i …
12
votes
Accepted
Unknown work of Nöbeling on topological/Hausdorff dimension
So, the sought for paper is:
Nöbeling, G., Hausdorffsche und mengentheoretische Dimension, Ergebnisse math. Kolloquium Wien 3, 24-25 (1931).
And here is a ``translation" (to English and to modern ma …
8
votes
1
answer
498
views
Bounding an "integral" from below by the Hausdorff measure of the domain
Let $(X,d)$ be an arbitrary metric space and $E \subset X$ also arbitrary. Fix $s \in (0,\infty)$.
Is it true that for any $ \delta > 0 $ and any collection of pairs $\{(A_i,a_i)\}_{i \in \math …
5
votes
Generalization of area and coarea formula for fractional Hausdorff measures
There are several intersting consequences of this abstract viewpoint about these formulas. I also would have liked to discuss why the coarea inequality is the backbone of the coarea formula, but let m …
2
votes
Coarea inequality, Eilenberg inequality
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 …
2
votes
Existence of subset with given Hausdorff dimension
The following is Corollary 7 of [1].
Theorem: For $X$ (an analytic subset of) a complete separable metric space, and $ s \in [0,\infty)$, the following is true about the Hausdorff measure $\mathcal{H …
0
votes
Accepted
A Curved/Warped Version of Fubini's Theorem
I have the answer here: Fubini's Theorem on Arbitrary Foliations
$$\int_U f = \int_{U_{\eta_0}} \left(\int_{U_\xi} f(\xi,\eta) \frac{|\det DG_{U_\xi} (\xi,\eta)| \cdot |\det DG_{U_{\eta_0}} (\xi,\eta …
6
votes
1
answer
1k
views
Fubini's theorem on arbitrary foliations
In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of …
2
votes
3
answers
773
views
A Curved/Warped Version of Fubini's Theorem
I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed …
6
votes
1
answer
312
views
Is Hausdorff Measure equal to Hausdorff Content on rectifiable (metric) spaces?
Let $(X,d)$ be an $\mathcal{H}^n$-rectifiable metric space, i.e. there exits a collection of Lipschitz maps from measurable subsets of $\mathbb{R}^n$ to $X$ such that $ \mathcal{H}^n(X \backslash \cup …