All Questions
Tagged with hausdorff-dimension geometric-measure-theory
32 questions
8
votes
1
answer
344
views
How large can the set of turbulent points be?
This question resisted attempts on MSE.
Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold:
$$\...
- 6,313
4
votes
1
answer
259
views
Hausdorff dimension of the zero set of the gradient of an eikonal function
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere with respect to Lebesgue measure.
What is the supremal Hausdorff dimension of the set on which $f$ is ...
- 6,313
5
votes
2
answers
248
views
Hausdorff dimension of the zero set of $\nabla f$
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $\nabla f$ nonzero almost everywhere with respect to Lebesgue measure.
What is the supremal Hausdorff dimension of the set on which $f$ ...
- 6,313
6
votes
0
answers
822
views
Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$.
Let $\Omega$...
- 6,313
1
vote
0
answers
98
views
Lower bound estimate for the sum $\sum \text{diam}(U)^d$ over all countable covers of a cube
This question is inspired from the definition of Hausdorff measure. Let $C$ be a closed unit hypercube in $\mathbb R^d$ (side length equal to one, including boundary. The cube itself is at top ...
- 1,565
4
votes
1
answer
196
views
On the set on which $|Df|$ is maximal for Lipschitz $f$
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz continuous function with strict Lipschitz constant $L > 0$.
That is, $|f(x) - f(y)| < L|x - y|$ for all $x \neq y$ in $\mathbb R^d$.
Question: ...
- 6,313
1
vote
0
answers
135
views
Hausdorff dimension of a compact Lie group [closed]
Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant Riemannian metric and let $m$ be left-invariant Haar measure on $G$.
Now that $G$ is a metric space ...
- 323
2
votes
0
answers
187
views
Relationship between Hausdorff dimension and covering number
Let $(X,d)$ be a compact metric space and recall that the $\epsilon$-external covering number $\mathcal{N}^{\epsilon}(X)$ of $X$ is defined by:
$$
\mathcal{N}^{\epsilon}(X) := \inf\left\{
N\in \mathbb{...
- 5,405
8
votes
1
answer
866
views
Fubini's theorem for Hausdorff measures
$B\subset \mathbb{R}^2$ is a Borel set. Define the slices $B_x:= \{y \in \mathbb{R}: (x,y) \in B \}$.
If $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$, presentations of Fubini's theorem often ...
- 675
7
votes
3
answers
548
views
Maximal Hausdorff dimension of the set on which derivatives do not agree
Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the supremal Hausdorff dimension of the set on which $f$ and $g$ are both ...
- 6,313
4
votes
1
answer
903
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: ...
- 649
3
votes
1
answer
151
views
For $\mathcal{L}^1$-a.e. $t\in\mathbf R$, is $n-1$ the Hausdorff dimension of level sets of a locally Lipschitz function $f:\mathbf R^n\to\mathbf R$?
Let $f:\mathbf R^n\to\mathbf R$ be a locally Lipchitz function. Denote $\mathrm H^n$ the $n$-dimensional Hausdorff measure. We know that for any $\mathrm H^n$-measurable subset $A\subset\mathbf R^n$, ...
8
votes
1
answer
213
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}^...
- 12.7k
6
votes
1
answer
205
views
Subspaces of metric spaces having prescribed dimension
Let $(X,d)$ be a metric space having Hausdorff dimension $\alpha>0$ and let $0<\beta<\alpha$. Is there a metric subspace of $X$ having Hausdorff dimension $\beta$?
- 325
3
votes
1
answer
230
views
Hausdorff dimension and $W^{1,1}$ functions
What can be said about the Hausdorff dimension of the image of a set by a $W^{1,1}$ map?
In other words,
what is the relationship between
$\mathrm{dim}_H f(A)$ and $\mathrm{dim}_H A$, where $f \in ...
- 839
7
votes
1
answer
272
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
...
- 1,045
23
votes
3
answers
1k
views
Existence of subset with given Hausdorff dimension
Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension.
For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff ...
- 1,045
11
votes
1
answer
962
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 $$
\...
- 28k
6
votes
3
answers
541
views
About the Hausdorff dimension of removable singularities of PDE
There are some interesting phenomenons about removable singularities (or extension problems).
In the theory of functions of several complex variables, we know the classical Hartogs theorem:
Let $f$ ...
- 181
1
vote
0
answers
251
views
Compact sets of Hausdorff dimension zero
I have a question about Hausdorff dimension. Suppose S is a compact subset of $\mathbb{R}^n$ whose Hausdorff dimension is zero. Does it follow that S can be covered by a finite DISJOINT union of ...
- 11
9
votes
1
answer
637
views
Is there a characterization of the Hausdorff measures?
It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
- 1,035
6
votes
5
answers
1k
views
Fractals of dimension zero
Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0?
I can think of something ad hoc like a Cantor middle $\frac13$ set where the ...
- 24.8k
3
votes
0
answers
204
views
Product Fractals
Here is a theorem found in the Falconer's book on fractal geometry:
Theorem: For any sets $E\subset \mathbb{R}^n$ and $F\subset \mathbb{R}^m$
$$
\dim_HF+\dim_HE\leq \dim_H(E\times F)\leq \dim_HE+\...
user11178
23
votes
3
answers
872
views
Best Hölder exponents of surjective maps from the unit square to the unit cube
The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...
- 60.6k
7
votes
3
answers
679
views
How can dimension depend on the point?
Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
- 8,086
6
votes
1
answer
670
views
Calculate Hausdorff measure with Frostman measures
Fix a metrix space $(X,d)$ and consider the Hausdorff (outer) measures $\mathcal{H}^s$ on $X$.
A Frostman measure on $X$ is a finite Borel measure $\mu$ such that there exists $C,t,r_0>0$ with $\...
- 9,650
7
votes
1
answer
2k
views
The relation between Hausdorff dimension of an $n$-manifold and $n$
It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general.
For the case of manifolds, suppose $M$ is a $n$-manifold with a metric(distance)...
- 285
11
votes
1
answer
1k
views
Geometric measures different from Hausdorff
$\newcommand{\RR}{\mathbb{R}}\newcommand{\calF}{\mathcal{F}}\newcommand{\diam}{\mathrm{diam}}$
In geometric measure theory there are various notions of $m$-dimensional measure for sets $A\subset \RR^n$...
- 12.7k
6
votes
2
answers
483
views
Can Hausdorff dimension make sets into a Tropical Semiring?
If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure
\begin{equation}
H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : \...
- 1,134
0
votes
2
answers
298
views
Hausdorff measure of the zero set
Let $f : \mathbb R^n\to \mathbb R$ continuous, for which there exist $x,y\in\mathbb R^n$, such that $f(x)f(y)<0$.
Is it true that the Hausdorff dimension of the zero set of $f$ is at least $n-1$?
- 2,933
3
votes
1
answer
581
views
Are there any good techniques for calculating Hausdorff measure?
I'm aware that many techniques have been developed for the purpose of calculating Hausdorff dimension (although I'm fairly unfamiliar with them), but my question is whether or not we have any good ...
- 173
6
votes
2
answers
2k
views
Hausdorff dimension of convex set in ${\bf R}^n$
I want to know the smoothness of convex set in ${\bf R}^n$.
Recall the following definition.
Definition : $X$ is a bounded closed convex set in ${\bf R}^n$ if for $x$, $y\in X$, the any $d$-...
- 1,100