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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2
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1answer
93 views

Estimate on the Hausdorff dimension of boundary of balls

I am reading Evans and Gariepy's book on GMT and I have a couple questions: 1) if E is a set of locally finite perimeter, is it true that E is $ \| \partial E\|$- measurable? 2) At a certain point, ...
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1answer
224 views

Hausdorff dimension of a smooth manifold

I read a book about fractal stating that without proof: every $m$-dimension $(m<n)$ smooth manifold $M$ in $\mathbb{R}^n$ has Hausdorff dimension $m$. How can we prove it?
4
votes
1answer
176 views

The approximating Hausdorff measure is not Borel

This is an exercise taken from Mattila, Geometry of sets and measures in Euclidean space, chapter 4. Exercise. Let $U$ be an open ball in $\mathbb{R}^n$ ($n\ge 2$) such that $d(U)=\delta$ [here ...
3
votes
1answer
79 views

Reference for an integral formula

Good morning, I'm reading a paper of W. Stoll in which the author uses some implicit facts (i.e. he states them without proofs and references) in measure theory. So I would like to ask the following ...
10
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1answer
575 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
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1answer
117 views

Volume is a Continuous Function

I am working on the following problem: Suppose $C \subset \mathbb{R}^d$ is a compact and non-empty set. Let $C_0 = C$ and let $C_t = \{x \in \mathbb{R}^d : d(x,C) \leq t \}$ for all $t >0$. ...
1
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1answer
156 views

Hausdorff measure of $n$-dimensional cube

This is a home work problem that I am stuck on even though it feels like it should be easy: Show that the n-dimensional Hausdorff measure of an $n$-dimensional cube is positive and finite. I can ...
0
votes
1answer
41 views

The figure of convex set in $S^{n-1}(1)$

If $X$ is a connected convex set in $S^{n-1}(1)$, then what is $\partial X$ ? Recall the following definition. Definition : $X$ is a convex subset of $S^{n-1}(1)$ if any two points in $X$ can be ...
5
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0answers
75 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
1
vote
1answer
51 views

Connection between local dimension and Hausdorff measures

Given that the local dimension of a measure $\nu$ at a point $w$ is defined by $$d[\nu](w) = \limsup_{r \to 0} \frac{\log(B(w,r))}{\log r}$$ is it possible to find a measure which has local dimension ...
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0answers
89 views

How Wiener Measure on $F(C([0,T]))$ is a Gaussian Measure

I'm looking for some simple proofs for the fact that on $(C[0,T],F(C([0,T])),P_{*})$ where $F$ represents Borel Sigma algebra , $P_{*}$ the Wiener Measure , then how to proove that $P_{*}$ measure is ...
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1answer
86 views

Limit Volume of Parallel Sets

Given $F \subset \mathbb R^n$ non empty and $\epsilon > 0$. Let $F_\epsilon$ be $\epsilon$-parallel set of $F$, $$F_\epsilon := \{x \in \mathbb R^n:d(x,F)\le\epsilon\},$$ with $d(x,F):= \inf_{y\in ...
0
votes
1answer
107 views

Is this map on a metric space upper-semi continuous?

Let $(X,d)$ be a metric space and $\mu$ a probability measure. Let $f(x)=\mu(B(x,r))$. Is this map upper semi continuous? I have some other assumptions since this is part of a larger proof but I'm ...
8
votes
1answer
176 views

Hausdorff Dimension of Set of Measure Zero

It's clear that every $A \subset \mathbb R^n $ with $\dim_H(A) < n$ we have $\mathcal H^n(A) = 0$. Is there any $A \subset \mathbb R^n $ with $\mathcal H^n(A) = 0$ but $\dim_H(A) = n$? Thank you.
20
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1answer
441 views

Measure theoretic definition of curl

Is there a good measure theoretic definition of curl? To give an idea of the sort of equation that I'm looking for, here's now I define grad and div. For the gradient, say we are given a Fréchet ...
3
votes
0answers
150 views

Fourier dimension of a measure restricted to an open set

Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that $\beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\}$. The Fourier ...
2
votes
1answer
173 views

Hausdorff Measure- Lower semi-continuity

By definition , when we are given a set $A \in \mathbb{R}^n$ , $$ H_\delta^{n-1} (\partial A ) = \inf \left\{ \sum_{j=1}^{\infty} \alpha_{n-1}\frac{1}{2^{n-1}} [\operatorname{diam}(U_j)] ^{n-1} ...
2
votes
2answers
59 views

How smooth is the distribution function of a convex polynomial?

Here is a prototype of the problem I have in mind: Let $P:\mathbb{R}^2\rightarrow\mathbb{R}$ be a strictly convex, nonnegative polynomial such that $P(0,0)=0$. Let $\alpha\geq 0$, and consider the ...
22
votes
1answer
615 views

Which sets are removable for holomorphic functions?

[Note: I received a version of this question via email and decided to answer it on MSE, where it might be useful to others.] Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some ...
2
votes
1answer
137 views

Equality in the Isoperimetric Inequality

Stein and Shakarchi, in their book Real Analysis, the third volume of the Princeton Lectures in Analysis series, give a proof of the isoperimetric inequality for closed rectifiable curves in ...
3
votes
1answer
92 views

Convolution square root of a Frostman measure

A probability measure $\mu$ on $\mathbb{R}^d$ is said to be a Frostman measure if $$\mu(B)\lesssim r(B)^\alpha \ \ \ \ (1)$$ for all open ball $B$, where $r(B)$ denotes the radius and $\alpha>0$. ...
7
votes
0answers
116 views

Higher Order Coarea Formula

I was wondering, if there is a generalization of the coarea formula to higher order derivatives, which would allow one, for example, to relate the norm of the Hessian of a real-valued function $u$ to ...
1
vote
2answers
89 views

Is critical Haudorff measure a Frostman measure?

Let $K$ be a compact set in $\mathbb{R}^d$ of Hausdorff dimension $\alpha<d$, $H_\alpha(\cdot)$ the $\alpha$-dimensional Hausdorff measure. If $0<H_\alpha(K)<\infty$, is it necessarily true ...
3
votes
0answers
134 views

Integral Geometry Reference Request

I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to ...
1
vote
1answer
336 views

Estimate the surface area of a 2D shape where the only known value is the length of the enclosing boundary

Wondering if it is possible to estimate the surface area of a 2D shape where the only known value is the length of the enclosing boundary, and that it is know the internal surface area is solid. ...
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0answers
255 views

Corollary of Lebesgue decomposition theorem and counter-example

Refferring to the Lebesgue decomposition theorem in Lebesgue decomposition theorem and fundamental theorem of calculus there is a corollary when the measure is the Lebesgue measure that states: if ...
1
vote
1answer
191 views

Surface measure for Lipschitz domain

Let $D\subset R^d$ be a bounded Lipschitz domain. Must there exist a bounded function $\Phi$ on $\partial D$ and collections of subsets $(\partial D )^{\epsilon} \subset \partial D $ (indexed by ...
3
votes
0answers
116 views

Radon-Nikodym derivative of the Hausdorff measure transform

Let $H^{m}$ be the $m$-dimensional Hausdorff measure, let $D = \operatorname{diag }(d_1,...,d_k)$ be a nonsingular scale matrix. Consider the change of measure formula: $$ \int\limits_{A} f(Dx) \; ...
6
votes
1answer
295 views

Is the Hausdorff outer measure regular?

An outer measure $\mu^*$ is said to be regular if for every set $A \subset X$ $$\mu^\ast (A)=\inf\{\mu^*(E) : E\supset A \text{ is } \mu^\ast\text{-measurable} \}$$ To check that an outer ...
0
votes
1answer
112 views

The modulus of curve family

I'm reading book "Geometric function theory and non-linear analysis" and there is one thing that I did not understand. A curve family $\Gamma$ is a collection of curves. An admissible density is ...
0
votes
1answer
110 views

Dimension invariance

Where can I find the Theorem of invariance of the dimension with diffeomorphisms? And about Between Homeomorphism? I also want to know about the Hausdorff dimension invariance what is deeper!
3
votes
0answers
165 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 ...
1
vote
0answers
81 views

Isoperimetric inequalities with relative perimeter

It is a well known result that if $\Omega\subset \Bbb{R}^N$ is an open set, with regular boundary (smooth, or Lipschitz) then the problem $$ \min_{E \subset \Omega |E|=c} Per(E)$$ has a solution, ...
0
votes
1answer
107 views

Reference for Radon measure

Can anyone provide references for Radon measures, Bounded variation functions space, and Lebesgue differentiation theorem for Radon Measures, Hausdorff dimensions? Note: Online references will be ...
3
votes
3answers
323 views

Total variation of (weakly) differentiable functions

the total variation of a function $u\in L^1(\Omega)$, $\Omega\subset \mathbb{R}^n$, can be defined as $$ \sup \{ \int_\Omega u \; \mathrm{div} g \; dx:\; g \in C_c^1(\Omega,\mathbb{R}^n), \; \lvert ...
5
votes
2answers
261 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 $$ \int\limits_{T(M)} f(x) H^{m}(dx) = |\det{T}| \int\limits_{M} f(T(x)) H^{m}(dx) $$ ...
1
vote
1answer
227 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 $$ ...
8
votes
2answers
552 views

Relations between various definitions of a Radon measure

The following various definitions of a Radon measure seem to be given for the Borel sigma algebra of different types of topological spaces: general, Hausdorff, locally compact, or locally compact ...
10
votes
4answers
368 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 ...
2
votes
1answer
283 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: ...
4
votes
1answer
324 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 ...
9
votes
1answer
445 views

Uncountable sets of Hausdorff dimension zero

Let $A \subset \mathbb{R}$ be a countable set. It is easy to see that $A$ has Hausdorff dimension $\dim_H(A) = 0$. Do there exist uncountable sets $A \subset \mathbb{R}$ with $\dim_H(A) = 0$?
6
votes
2answers
160 views

Methods to define areas

I always thought that areas are defined by integrals, until I read Michael Spivak's Calculus p.289: The desire to define area was the motivation, both in this book and historically, for the ...
2
votes
3answers
95 views

Hausdorff measure with non-power test function

At my analysis course some time ago we were told that there is definition of Hausdorff measure through the test functions which are continuous and non-decreasing $h:(0,\infty)\to(0,\infty)$ and ...
4
votes
1answer
343 views

How to understand currents in geometric measure theory?

I find it is hard to catch the current, sometimes it is just the picture as its support set (if I do not miss it). What is the heart idea of the current? What are the benefits to introduce such an odd ...
6
votes
0answers
261 views

Ham sandwich theorem for integrable functions?

The classical ham sandwich theorem says that given $n$ measurable sets in $\mathbb{R}^n$, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single $(n − ...
6
votes
2answers
253 views

Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem

A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume". At that time for me to write ...
2
votes
3answers
330 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,roughly ...
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0answers
597 views

How do we prove that a sphere maximizes the volume enclosed among all simple closed surfaces of given surface area?

How do we prove that among all closed surfaces with a given surface area, the sphere is the one that encloses the largest volume, and not do it by cases? so far I've tried is that I know the formula ...
3
votes
1answer
391 views

Calculating the upper Minkowski dimension of the set $\{0,1,\frac{1}{2}, \frac{1}{3}, \ldots \}$

The upper Minkowski dimension of a compact set $A$ in $\mathbb{R}$ is defined as $$ \overline{\dim}_M = \inf \{ \epsilon > 0 : \text{ there is a constant } C(\epsilon) \text{ such that } ...