The study of the geometric structure of measures, as well as the study of geometry from a measure-theoretic viewpoint, geometric measure theory has applications in partial differential equations, harmonic analysis, differential and Riemannian geometry, as well as calculus of variations. Statements ...

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26
votes
1answer
801 views

Which sets are removable for holomorphic functions?

Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some class of functions from $\Omega$ to $\mathbb C$. A set $E\subset \Omega$ is called removable for holomorphic functions of class ...
13
votes
7answers
655 views

Has the notion of having a complex amount of dimensions ever been described? And what about negative dimensionality?

The notion of having a number $a \in \mathbb{R}_{\geq 0} $ associated to any metric space is described by the definition of a "Hausdorff Dimension". I was wondering if work has been done on spaces ...
32
votes
4answers
775 views

To show that the set point distant by 1 of a compact set has Lebesgue measure $0$

Could any one tell me how to solve this one? Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$ Show that $A$ has Lebesgue measure $0$. Thank you!
10
votes
1answer
784 views

Inner regularity property of Radon measures in metric spaces

Let us agree to say that $\mu$ is a Radon measure on a metric space $X$ if it is a Borel measure which is finite on compact subsets and is such that: Every measurable subset $A$ is outer regular, ...
4
votes
1answer
387 views

Lebesgue measure on normal matrices

Consider the space of $n\times n$ complex matrices, and equip it with its Lebesgue measure $dX$, seen as a $2n^2$-dimensional real vector space [edit: or better, a complex vector space (see the answer ...
30
votes
2answers
549 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
5
votes
1answer
374 views

Minkowski Content

Could someone provide some intuition behind the $n$-dimensional Minkowski Contentthe $n$-dimensional upper Minkowski Content of $\mathcal{A}$ as $$\mathfrak{M}^{*m} (\mathcal{A}) : = \lim_{\epsilon ...
3
votes
0answers
180 views

Co-area formula involving non-integer Hausdorff measure

Is there any co-area formula involving non-integer Hausdorff dimension? Moreover is it sensible to write the following: Let $S$ be a subset in $ \mathbb{R}^n$ with Hausdorff dimension s ...
0
votes
1answer
109 views

Proving ineqalities for the similarity dimension

a. Let $K$ be the attractor of the IFS $\{f_1,\dots f_n\}$ which satisfies SSC (i.e $f_i(K)\cap f_j(K)=\emptyset\forall i\neq j$) where for all $i, c_i$ such that $ 1\le i\le n, \space ...
0
votes
1answer
87 views

Hausdorff measure of the middle third Cantor set and Compactness

In the proof of the Hausdorff dimension of the middle third cantor set I cannot understand why we need the following underlined statement. I cannot understand why we need only consider closed ...
5
votes
2answers
302 views

Change of variable within an integral of the Hausdorff measure

Let $T \colon \mathbb{R}^n \to \mathbb{R}^n$ be a linear map, $H^{m}$ be a Hausdorff measure. Is it true that $$ \int\limits_{T(M)} f(x) H^{m}(dx) = |\det{T}| \int\limits_{M} f(T(x)) H^{m}(dx) $$ ...
4
votes
0answers
115 views

Limits to the growth of the volume of a union of spheres

Assume that $x_i$, $i=1,\ldots,m$ are points in $\mathbb{R}^n$, with the maximal distance between any two of them being at most $1$. Define $$ a(r)=\mu\Bigl(\bigcup_{i=1}^m B(x_i,r)\Bigr),$$ where ...
3
votes
1answer
313 views

Volume form and Hausdorff measure

Let $M$ be a smooth orientable $(n-1)$-dimensional submanifold in $\mathbb{R}^n$, $dS$ be its volume form and $dH^{n-1}(x)$ be an $(n-1)$-dimensional Hausdorff measure. How to show than that $$ ...
2
votes
1answer
263 views

The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$

I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
2
votes
1answer
296 views

Banach-Tarski Paradox on the middle third Cantor set

In analysis and topology, the middle third Cantor set $C$ is often a very interesting topic of discussion. My question is that is it possible to have some sort of measure preserving bijection $f: ...
2
votes
3answers
341 views

How can a density be larger than $1$?

From Frank Morgan: Geometric Measure Theory, Fourth Edition: A Beginner's Guide, page 13,the $2$-dimensional density of the cone $x^2+y^2=z^2$ at $0$ is $\sqrt{2}$. I feel strange of that´╝îroughly ...
1
vote
0answers
101 views

Von Neumann and Hausdorff continuous dimensions are related?

Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous ...
1
vote
1answer
314 views

metric and measure on the projective space

Let $RP^n$ be the $n$-dim real projective space. I have the following four questions. What is the so called standard metric on $RP^n$? More generally, consider a metric space $M$ with an equivalent ...
1
vote
1answer
507 views

Riemannian measure and Hausdorff measure in a general Riemannian Manifold

Let $ M $ be a Riemannian manifold and let $ \mu $ be its Riemannian measure. This is the measure obtained by Riesz reprersentation theorem such that for every continuous function with compact support ...
0
votes
1answer
41 views

Upper Minkowski content and finite Hausdorff measure

Does someone know an example of a set $E$ with positive finite $s$-Hausdorff measure, Minkowski dimension $s$, and infinite $s$-dimensional upper Minkowski content ? The $s$-dimensional upper ...