Questions tagged [hausdorff-measure]
Questions about Hausdorff measures, their variants (such as spherical Hausdorff measures) and generalisations.
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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}...
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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 ...
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Naïve definition of a measure on a fractal
This question was previously posted on MSE.
Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$.
One option would be to use ...
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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 ...
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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 ...
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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 ...
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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\...
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Convergence of the perimeter of level sets
I have already posted this question on Math StackExchange. Suppose you have a sequence of $C^1$ functions $\{\phi_n\}_{n\in \mathbb{N}}$ defined on $\mathbb{R}^n$ that converges in $C^{1}_{\mathrm{loc}...
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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 $...
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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$...
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The Hausdorff measure of intersection of annulus and conformal curve
Recently I came across a problem in my research. Let $g:[0,1]\to\mathbb{C}$ be a restriction of a conformal map that is defined in a simply connected domain $\Omega\subseteq\mathbb{C}$ that include $[...
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$\mathscr{H}^{n-2}(\Sigma)< \infty$ implies $\mathscr{H}^{n-1}(\pi(\Sigma))=0$
Let $\Sigma\subset \mathbb{R}^{n+1}$ be a set with $(n-2)$-dimensional Hausdorff measure finite, i.e. $\mathscr{H}^{n-2}(\Sigma)<\infty$. Let $\pi:\mathbb{R}^{n+1}\to \mathbb{R}^n$ be the ...
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Existence of an $\alpha$-regular measure with positive measure on a binary digits do not have a limiting frequency
let $$X=\left\{ \sum_{n=1}^{\infty}a_{n}2^{-n}:a_{n}\in\left\{ 0,1\right\} ,\liminf\frac{1}{n}\sum_{i=1}^{n}a_{i}<\limsup\frac{1}{n}\sum_{i=1}^{n}a_{i}\right\} $$
I'm studying fractal geometry and ...
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A counterexample to regular boundary points for minimizers of variational integrals under subquadratic growth
Let $\Omega\subset\mathbb{R}^n$ for some $n\geq 3$ be an open bounded set with at least Lipschitz boundary. Let $p\in (1, 2), N>1$ and $f: \overline{\Omega} \times\mathbb{R}^N\times\mathbb{R}^{Nn}\...
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Relationship between Hausdorff dimension and covering number
Let $(X,d)$ be a compact metric space and recall that the $\epsilon$-external covering number $\mathcal{N}^{\epsilon}(X)$ of $X$ is defined by:
$$
\mathcal{N}^{\epsilon}(X) := \inf\left\{
N\in \mathbb{...
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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 ...
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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 ...
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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
$$W(x,y)=\begin{bmatrix} 1/2 & 0 \\ 0 &...
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Cardinality of intersections of lines with irregular 1-sets in the plane
From Falconer's book (The geometry of fractal sets), Lemma 3.2 says that the intersection of irregular 1-sets with straight lines is of zero $H^1$ measure. What do we know about the cardinality of ...
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Are the extensions of the expected value, below, finite for all functions in only a shy subset of all measurable functions?
This is a follow up to this post, where I wish to verify whether one of the statements (in the post) is true but first let's recap the definitions:
Let $(X,d)$ be a metric space. If set $A\subseteq X$,...
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Finding a unique and finite expected value for almost all measurable functions?
Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the ...
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Lower bound estimate for the sum $\sum \text{diam}(U)^d$ over all countable covers of a cube
This question is inspired from the definition of Hausdorff measure. Let $C$ be a closed unit hypercube in $\mathbb R^d$ (side length equal to one, including boundary. The cube itself is at top ...
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How to show that this function is continuous (Geometric Measure Theory)
I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by
$$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$
is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...
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Using programming to measure the uniformity of measurable subsets of the unit square?
This is a follow up to this post using this answer:
Let $S:=[0,1]^2$ be the unit square. "Partition" $S$ naturally into four congruent squares $S_{1,j}$ (with side length $1/2$ each), where ...
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Continuous dependence of the (infinite) roots of a polynomial on its coefficients
I'm trying to show the continuous dependence of the roots of a polynomial on its coefficients when the root number can be infinite (e.g., $x-y$). I don't know much about algebraic geometry but after I ...
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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,\...
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Finding examples of functions which are infinite or undefined with current extensions of the expected value?
Preliminaries
Consider the expectations desribed in this paper, which is an extension of the Lebesgue density theorem; this paper which is an extension of the Hausdorff measure, using Hyperbolic ...
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