Skip to main content
Search type Search syntax
Tags [tag]
Exact "words here"
Author user:1234
user:me (yours)
Score score:3 (3+)
score:0 (none)
Answers answers:3 (3+)
answers:0 (none)
isaccepted:yes
hasaccepted:no
inquestion:1234
Views views:250
Code code:"if (foo != bar)"
Sections title:apples
body:"apples oranges"
URL url:"*.example.com"
Saves in:saves
Status closed:yes
duplicate:no
migrated:no
wiki:no
Types is:question
is:answer
Exclude -[tag]
-apples
For more details on advanced search visit our help page
Results tagged with
Search options not deleted user 91442

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 …
Behnam Esmayli's user avatar
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 …
Behnam Esmayli's user avatar
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 …
Behnam Esmayli's user avatar
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 …
Behnam Esmayli's user avatar
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 …
Behnam Esmayli's user avatar
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 …
Behnam Esmayli's user avatar
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 …
Behnam Esmayli's user avatar
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 …
Behnam Esmayli's user avatar
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 …
Behnam Esmayli's user avatar
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 …
Behnam Esmayli's user avatar
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 …
Behnam Esmayli's user avatar