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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10 views

Function/Measure Notation in Geometric Measure Theory

I'm trying to understand a formula of this kind $$ ...=\phi_\sharp \left ( f \mathcal{H}^n \right ) $$ where $\mathcal{H}^n$ is the n-dimensional Hausdorff measure on a measure space $X$, $\phi : X ...
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1answer
12 views

Derivative of volume of given set

As picture below ,how to compute the $\partial_t |\Omega_t|$ ? The picture below is from the 32 page of Maximum principles and the method of moving planes. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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34 views

Minimum of $F$ over Finite Perimeter Sets in $\mathbb R^N$

Problem: Let $G$ be a bounded Borel set. Let $X$ be the set of finite perimeter sets in $\mathbb R^N$ and $F: X \to \mathbb R \cup \{+\infty\}$ defined as \[ F(E)= \begin{cases} Per(E) \hspace{1,...
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26 views

Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

This question has been post in here. Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable). ...
3
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0answers
48 views

Constructing a null set and a Lipschitz function nowhere differentiable on it

I'm trying to solve the following exercise. Now, Rademacher's theorem says that locally Lipschitz functions are $\mathcal L^N$-a.e. differentiable, so $E$ must be a null set, and this is clearly ...
6
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389 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 $$ \int\limits_{T(M)} f(x) H^{m}(dx) = |\det{T}| \int\limits_{M} f(T(x)) H^{m}(dx) $$ ...
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249 views

Change of Variables for Hausdorff Measure

(Read bounty text for answering question) Let $H^{m}$ be the $m$-dimensional Hausdorff measure. Let $D$ be a linear transformation matrix. Consider the change of measure formula: $$ \int\limits_{A} ...
2
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1answer
35 views

$f : B \to \mathbb R^n$ such that $\mathcal L^n(N)=0 \implies \mathcal L^n(f(N))=0$

Let $B$ an open ball in $\mathbb R^n$. Let $f:B \to \mathbb R^n$ measurable and satisfy the property that $N \subset B, \mathcal L^n(N)=0 \implies \mathcal L^n(f(N))=0$, where $\mathcal L^n$ is the ...
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0answers
18 views

Help understanding proof of Frostman's Lemma - issue technical or termonological?

I was reading Hochman's proof of Frostman's lemma in his online lecture notes here and got hung up. I'm not sure if I'm missing a part of the proof or I'm misunderstanding the theorem itself. The ...
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1answer
56 views

Using small spherical balls to fill a cube and also find volume of the cube

Is it possible to calculate volume of a cube (with volume $L^3$) by filling it with small balls each with a radius $r_N$ and the balls are disjoint. Let the number of balls be $N$ $$\lim\limits_{N\...
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1answer
31 views

Two Borel disjoint sets such that the perimeter of union is less than the sum of perimeters

Exercise: Find two Borel disjoint and bounded sets $E, F \subset \mathbb R^n$ such that $\operatorname{Per}(E) + \operatorname{Per}(F) > \operatorname{Per}(E \cup F)$. ($\operatorname{Per}(A)$ is ...
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38 views

Perimeter of the Unit Ball in $\mathbb R^n$

Exercise: Calculate the perimeter of the unit ball in $\mathbb R^n$, i.e. show that $\mathcal H^{n-1}(S^{n-1})=n\omega_n$, where $\mathcal H^{n-1}$ is the Hausdorff measure of dimension $n-1$ and $\...
12
votes
1answer
324 views

Surface area from indicator function

I know that the volume and the surface area of a sphere of radius $R$ are related by a derivative: $$V(R)=\frac{4}{3}\pi R^3$$ $$A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$ I am asking if an ...
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128 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 ...
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1answer
33 views

Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$.

Problem: Let $f: [0,1] \to \mathbb R$ and $\Gamma(f)$ its graphic. Show that $\mathcal H^1(\Gamma(f)) \geq 1$. Attempt: Well, if $f \equiv 0$ we get 1. Provided some sort of goodness like $f \in C^1$ ...
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2answers
232 views

Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of $H$’s and $T$’s. Let $N$ denote the number of tosses until you see “$TH$” for the first time. For example, for the sequence $HTTTTHHTHT$, we needed $...
3
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0answers
339 views

Is a Jordan curve encircling a finite-perimeter set rectifiable?

Let $\gamma:[0,1]\rightarrow \mathbb R^2$ be a (continuous) simple closed curve (Jordan curve). The curve is not assumed to be rectifiable, i.e. we don't assume a priori that the length of the curve $$...
0
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1answer
49 views

Image of Lipschitz map measure zero

Let $f\colon\mathbb{R}^{n-1}\to\mathbb{R}^n$ be Lipschitz, i.e., $|f(x)-f(y)|\leq C|x-y|$ for some $C>0$. How do we show that the image $f(\mathbb{R}^{n-1})$ has Lebesgue measure zero? I can see ...
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1answer
356 views

Hausdorff Dimension of a Smooth Manifold

I read a book about fractals which stated without proof: Every $m$-dimensional $(m<n)$ smooth manifold $M$ in $\mathbb{R}^n$ has Hausdorff dimension $m$. How can we prove this?
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126 views

Can you determine the length of a curve by the lengths of its projections onto planes?

If $\,\Gamma \subset \mathbb R^n$ is $1$–rectifiable, then its Hausdorff measure is equal to its integral geometric measure. That is, $$\displaystyle\mathcal H^1\left(\Gamma\right) = \int_{G\left({1,\...
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vote
1answer
45 views

Are every measure on $\mathbb{R}^{n}$ borel and/or regular?

I saw the above question in an exam paper and I am not sure how to even start. The question is true for the case of Lebesgue measure but I am not sure for arbitrary measures. I have tried looking at ...
3
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1answer
112 views

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ submanifold....
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1answer
80 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
3
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3answers
255 views

Online reference about Geometric Measure Theory.

I would like to find an online reference about the basics of Geometric Measure Theory. The reference should treat such things as regions and isoperimetric surfaces. Can you tell me, where I can find ...
4
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1answer
103 views

Any two polish atomless probability spaces are isomorphic

On page 7 of Villani's Optimal Transport: Old and New (page 19 in this preprint), he states that any two polish atomless probability spaces $(\mathcal{X},\mu)$ and $(\mathcal{Y},\nu)$ are measure-...
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1answer
361 views

Radon-Nikodým derivative with respect to the Lebesgue/Hausdorff measure is always defined

Question 1. Is it possible to say that the Radon-Nikodým derivative of locally-finite Borel measure on $\mathbb R^n$ with respect to the Lebesgue measure is always defined but may be a generalized ...
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16 views

Lower semi continuity for the norm of the speed

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $\gamma_n: [0,1]\to \overline{\Omega}$ be a sequence of Lipschitz functions which converges uniformly on $[0,1]$ to a Lipshitz ...
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0answers
17 views

Bounding below Hausdorff measure of conected set

I'm trying to prove that for every connected set $E\subset\mathbb{R}$, $H^1(E)$, the Hausdorff measure is bounded below by $\text{diam}(E)$. In the answer there, it was suggested to use a Lipschitz ...
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1answer
18 views

Can we bound a Brenier map between uniform distributions with the Hausdorff distance between their supports?

Let $A,B$ be compact subsets of $\mathbb{R}^n$. Let $\mu_A$ (resp. $\mu_B$) be uniform probability measures over $A$ (resp. $B$). Then as a consequence of Brenier's theorem there is a one-to-one ...
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0answers
18 views

Identification of the normal cycle to a closed curve

I'm studying the normal cycles in Morvan's book "Generalized curvatures"; He says that the normal cycle of a domain D bounded by a smooth curve is the current ( i.e the linear continuous functional ...
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26 views

Geometrical interpretation of complex exponential integral

Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ (...
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20 views

Existence of a locally essentially unbounded integrable function

Does there exist an integrable function $f\colon [0,1]\to \mathbb{R}_+$ such that for every $0\leq a < b\leq 1$ we have $\| \chi_{(a,b)} f\|_\infty = + \infty$?
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2answers
244 views

How to correctly calculate the fractal dimension of a finite set of points?

The box-counting dimension is defined by: $\lim\limits_{\epsilon \to 0} \dfrac{N(\epsilon)}{1/ \epsilon}$ What works well if you are solving algebraically or if you can recursively generate more ...
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2answers
33 views

Non-measurable subset of a null set.

I am reading measure theory,and I am searching an example in which a measurable null set have a non-measurable subset because this is the reason that,s why we are studying about complete measure....
0
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1answer
15 views

Characterizing isotropic measures

A Borel measure $\mu$ on $S^{n-1}$ is called isotropic if $$\int_{S^{n-1}} \langle \theta, x \rangle^2 d\mu(x)=\frac{\mu\left({S^{n-1}}\right)}{n}$$ for all $\theta\in S^{n-1}$. This means that in ...
0
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1answer
29 views

definition of $f$ being measurable

I am having troubles understanding the definition of a function being measurable. I have that for a measure space $(\Omega, \mathcal{F}, \mu)$ a function $f: \Omega \to \mathbb{R}$ is measurable if $\{...
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0answers
9 views

Uniform random samples inside bounded region

In an $n$ dimensional space I have a region bounded by pairs of hyperplanes: \begin{equation} b_j \le \sum_{i=1}^n a_{ij} x_i \le c_j, \quad\forall j=1,\ldots,m. \end{equation} We can include in those ...
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1answer
21 views

Use uniform convergence to control the derivative of function

Let $f_n$: $I:=[-1,1]\to \mathbb R$ be given, and we also assume that $f_n\in C^\infty(I)$. Assume $f_n\to 0$ uniformly in $I$, and we know $$ \limsup_{n\to\infty}\int_I\sqrt{1+(f'_n)^2}<\infty $$ ...
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1answer
70 views

Generalizing Green's Theorem

Let $\phi:[0,1] \rightarrow \mathbb{R}^2$, with $\phi(t)=(x(t),y(t))$, a function satisfying the following assumptions: (i) $x(t)$ and $y(t)$ are absolutely continuous; (ii) $\phi(0)=\phi(1)$, the ...
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67 views

Pointwise convergence of a sequence of approximate limits of BV functions.

So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...
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1answer
33 views

The level set of Sobolev function

Let $u\in W^{1,2}(\Omega)$ where $\Omega\subset \mathbb R^2$ is open bounded, smooth boundary. Moreover, we have $0\leq u\leq 1$. Let set $\Gamma$ be defined as $$ \Gamma:=\{x\in\Omega,\,\, u(x)=0\} $...
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2answers
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Lipschitz continuous one-to-one mapping from subset $K\subset\mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Let $f:\mathbb{R}^n\to \mathbb{R}^{n-1}$ and $K\subseteq \mathbb{R}^n$ be a set of positive Lebesgue measure. What kind of regularity do we have to impose on $f$ (e.g., $C^1$, Lipschitz) to conclude ...
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approximate $C^1$ function by Holder function

I am trying to prove the following Lemma: Suppose $u$: $\mathcal R^N\to \mathbb R$ is $C^1$. Then for each $\epsilon>0$, there exists a $C^{1,\alpha}$ function $\tilde f$ such that $$ \...
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2answers
45 views

Example of a compact non-$G_{\delta}$ set?

For a homework assignment, I recently solved the following problem in Folland's real analysis text. Folland's Question: If $X$ is a Locally Compact Hausdorff space, and $K \subset X$ is a compact $...
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2answers
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A formula for Perspective measurement

If I have a photograph of a rectangular object whose image recedes with perspective, is it possible to work out the scale of measurements along that object? For example: I have a photograph of a (...
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24 views

Topologies on Spaces of k-rectifiable sets, and $C^k$ convergence

I've been thinking about topologies on the spaces of k-rectifiable sets (Hausdorff metric topology, varifold topology, etc) in $\mathbb{R}^n$, and I'm wondering if there are hypotheses under which ...
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22 views

Estimating Self-Convolution of Surface Measure on Sphere

Let $n\geq 2$, and let $\sigma$ denote the standard surface measure on the $n$-sphere $S^{n-1}$, normalized to have total measure $1$. According to the exercise at the end of Terence Tao's blog post ...
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22 views

n-1 dimensionnal Hausdorff measure and codimension 1 measure

I've been told that on a n-dimensionnal Riemannian manifold, the Hausdorff measure of dimension n-1 and the codimension 1 measure $v_{-1}$ (defined below) are mutually absolutely continuous. I've ...
7
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1answer
190 views

Uniform estimate for the boundary area of the union of closed unit balls

Let $A$ be the union of closed unit balls around countable many points $\{v_i\}\in\mathbb R^n$. If $\{v_i\}$ are chosen from a fixed unit ball, is it true that the boundary of $A$ has area $< c(n)$?...
5
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254 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 \begin{equation*} \beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq C(1+|x|)^{-\gamma/2}\right\...