Questions tagged [hausdorff-measure]

Questions about Hausdorff measures, their variants (such as spherical Hausdorff measures) and generalisations.

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4
votes
3answers
196 views

Recovering the length metric from Hausdorff measure

The metric cannot be recovered from its Hausdorff measure in general. Now, assume that $(X,d_X)$ and $(Y, d_Y)$ are connected compact length spaces and induce $n$-dimensional Hausdorff measures $\...
2
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0answers
93 views

Green's identity with a different norm

Let $\Omega \subset \mathbb{R}^n$ be a domain with a smooth boundary $\Gamma$. Suppose that $f, g \colon \mathbb{R}^n \to \mathbb{R}$ are of class $C^\infty( \overline{\Omega})$. Then Green's first ...
4
votes
2answers
191 views

Hausdorff dimension of Julia set

Can anyone show me the proof "Hausdorff dimension of Julia set is strictly positive"? For purpose to prove this we might have to prove the green function of basin of attraction to infinity ...
2
votes
1answer
120 views

Hausdorff dimension and surface measure

Could someone please indicate me some reference that contains the proof of the following theorem? Below $\mathcal{H}^n$ denotes the $n$-dimensional Hausdorff outer measure in $\mathbb{R}^n$. Theorem: ...
3
votes
1answer
142 views

Average of the sum of dirac measures

Let $(M^n,g)$ be a closed smooth Riemannian manifold. Consider a set $\mathcal B_{\epsilon}$ which consists of a maximal number of points in $M$ with pairwise distance no smaller than $\epsilon$. We ...
2
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0answers
117 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 ...
1
vote
1answer
233 views

Why is the Hausdorff measure of this set zero?

Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set, and let $\phi:\Omega\to\mathbb{R}^N$ be a $C^1$ function with the property that $\phi^{-1}(0)\neq\emptyset$, and $\nabla\phi(x)\neq 0,\ \...
0
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0answers
69 views

Signed distance function

Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with uniform Lipschitz boundary. Consider the signed distance function: $d:\mathbb{R}^N\to\mathbb{R},\ d(x)=\begin{cases} \mathrm{dist}(x,\...
5
votes
1answer
150 views

The product of two Hausdorff measures

Let $(X,d_X)$ and $(Y,d_Y)$ be two compact metric space with Hausdorff dimensions $\dim_H(X)=n$ and $\dim_H(Y)=m$ and Hausdorff measures $\mathcal{H}^{n}$ and $\mathcal{H}^{m}$. Assume that $\dim_H(...
0
votes
0answers
89 views

2 dimensional Hausdorff measure and area measure on the hyperbolic plane

$\newcommand{\diam}{\operatorname{diam}}$From the Encyclopedia of Mathematics, the Hausdorff measure on a generic metric space (X,d) can be defined using $H^\alpha_\delta (E):=\omega_\alpha \inf \{\...
0
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0answers
31 views

$2$-dimensional density of the cone

I'm reading Morgan's Geometric Measure Theory: A Beginner's Guide and he says that the $m$-dimensional density of a set $A$ is given by $$ \begin{equation} \Theta^m (A,a) = \lim_{r \rightarrow 0} \...
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0answers
34 views

Hausdorff dimension of cookie-cutter sets using topological pressure

I'm reading the proof of Theorem 5.3 from "Techniques in Fractal Geometry" by Kenneth Falconer - the theorem wich gives the Hausdorff dimension of a cookie-cutter set using topological ...
1
vote
1answer
170 views

Finding the dimension of the intersection of two real algebraic varieties

Suppose we have two polynomials $p, q \in \mathbb{R}^3$ and we are interested in their simultaneous zeros. Parameter counting tells us that the zero set most probably is going to be a one dimensional ...
4
votes
0answers
114 views

Continuity of the Lebesgue measure w.r.t the Hausdorff metric

I have a question linked to Interplay of Hausdorff metric and Lebesgue measure. Let us denote as $\mathcal K(\mathbb R^n)$ the space of compact subsets of $\mathbb R^n$ endowed with the Hausdorff ...
4
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0answers
160 views

Classification of Euclidean-invariant measures?

Is there a classification of measures on $\mathbb R^n$ which are invariant under (Euclidean) isometries? Hausdorff measures of all kinds are examples -- could that be all of them? More precisely, By ...
1
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0answers
58 views

Continuity haussdorff measure w.r.t level set and coarea formula

Let $M$ be a smooth compact riemannian manifold of dimension $n$ without boundary, and $A$ a measurable subset of $M$. Let $f:M\mapsto\mathbb{R}$ be a $C^1$ function and $\varphi:\mathbb{R}\mapsto\...
6
votes
1answer
295 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 \...
4
votes
1answer
114 views

Can passing to a length metric increase Hausdorff measure?

For concreteness, let's say that $(X,d)$ is a metric space homeomorphic to $\mathbb{R}^2$ whose Hausdorff 2-measure $\mathcal{H}_d^2$ is locally finite. We can pass from $(X,d)$ to the length metric, ...
6
votes
1answer
192 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 \...
5
votes
0answers
88 views

How do sets with unit fractional Hausdorff measure of dimension $>1$ look like?

Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask: Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^...
2
votes
1answer
226 views

Gromov-Hausdorff distance between weighted tree graphs

I would like to measure the similarity between a pair of weighted tree graphs. According to this post, this can be done by regarding the trees as metric spaces and then applying the Gromov-Hausdorff ...
5
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0answers
191 views

How to calculate the volume of a parallelepiped in a normed space?

Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
2
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0answers
77 views

Examples of essentially sub-linear functions

A dimension function is an increasing, continuous function $% f:\mathbb R_{+}\rightarrow \mathbb R_{+}$ such that $f(r)\to 0$ as $r\to 0$. Say that a dimension function $f$ is essentially sub-linear ...
7
votes
1answer
209 views

Hausdorff dimension of the boundary of fibres of Lipschitz maps

Let $f: \mathbb{R}^m\rightarrow \mathbb{R}^{m-k}$ be a Lipschitz map. Can we get a uniform estimate on the Hausdorff dimension of the boundaries of fibres of $f$? I.e. do we have an upper bound for ...
10
votes
1answer
643 views

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 $$ \...
1
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0answers
131 views

How to show that this function is continuous (Geometric Measure Theory)

I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by $$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$ is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...
7
votes
1answer
164 views

Examples of probability measures with `fake' decay

To be concise, I am wondering whether there are natural examples of probability measures $\mu$ compactly supported on the real line which satisfy $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ ...
14
votes
0answers
422 views

Isoperimetric inequality and geometric measure theory

The following version of the isoperimetric inequality can be easily deduced from the Brunn-Minkowski inequality: Theorem. If $K\subset\mathbb{R}^n$ is compact, then $$ |K|^{\frac{n-1}{n}}\leq n^{-1}...
3
votes
0answers
87 views

Converse on the rectifiability of products of rectifiable sets

Let $1\leq k\leq m$ and $1\leq l\leq n$ fixed integers, $\mathscr{H}^k$ the $k$ dimensional Hausdorff measure and $E\subset \mathbb{R}^m$. We say that : (1) $E$ is $k$ rectifiable if there exists $C\...
3
votes
0answers
216 views

When is Hausdorff measure locally finite?

Let $(X,d)$ be a metric space and let $H^\alpha$ denote the $\alpha$-dimensional Hausdorff Borel measure on $X$, where $\alpha$ is the Hausdorff dimension of $X$. Are there any simple conditions on $X$...
3
votes
1answer
117 views

volume of region between two manifolds

The question is motivated by a simple example: the area of a ring is $\pi(R^2-r^2)$, where $R$ and $r$ are the radii of the outer and inner circles respectively. Let $C$ be the 'middle circle' with ...
5
votes
1answer
472 views

Hausdorff approximating measures and Borel sets

Suppose $ 1 \leq m \leq n $ are integers and for each $ 0 < \delta < \infty $ let $\mathscr{H}^{m}_{\delta} $ be the size $ \delta $ approximating measure of the $ m $ dimensional Hausdorff ...
10
votes
2answers
2k views

Radon-Nikodym theorem for non-sigma finite measures

Let $(X,\mathcal M, \mu)$ be a measured space where $\mu$ is a positive measure. Let $\lambda$ be a complex measure on $(X,\mathcal M)$. When $\mu$ is sigma-finite, the Radon-Nikodym theorem provides ...
6
votes
1answer
417 views

Generalization of Area and Coarea formula for fractional Hausdorff measures

Let $X,Y$ be polish spaces, $s,t>0$ and $F:X\to Y$ locally Lipschitz continuous such that $X$ is $\sigma$-finite w.r.t. the $(s+t)$-dimensional Hausdorff measure $\mathcal{H}^{s+t}$. The Eilenberg ...
6
votes
2answers
163 views

Subsets $X$ such that their Hausdorff outer measure is not finite

Let $H^d:\mathcal{P}(\mathbf{R}^n) \to \mathbf{R}\cup \{\infty\}$ be the $d$-dimensional Hausdorff outer measure on $\mathbf{R}^n$, for some $0<d<n$ with $n$ integer, which is constructed in the ...
2
votes
1answer
554 views

Results for Hausdorff Measure after Linear Transformation

For the Sierpinski Triangle, $S$, the $d$ dimensional Hausdorff measure is given by, $H^{d}(S)$. If a linear transformation, $W$ is applied to $S$, with $$W(x,y)=\begin{bmatrix} 1/2 & 0 \\ 0 &...
11
votes
2answers
974 views

Are there any exact results for Hausdorff Measure?

The computation of the Hausdorff measure is extremely difficult due to the infimum appearing in its definition. This has made the calculation of the Hausdorff measure for nearly all fractals difficult ...
0
votes
1answer
393 views

One-dimensional Hausdorff measure of preimages

Let $\Omega$ be an open subset of $\mathbf{R}^n$. For a mapping $f: \Omega\to \bf{R}^n$, what kind of condition ensures that the one-dimensional Hausdorff measure of $f^{-1}(E)$ is zero whenever $E$ ...
14
votes
1answer
1k views

Hausdorff measure and the volume form

There are two tools, generalizing a concept of a volume to the case of submanifolds in $\mathbb{R}^n$, namely the Hausdorff measure $H^k$ and the volume form. The question is how to show that if $M$ ...