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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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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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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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). ...
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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 ...
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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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+100

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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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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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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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 $\...
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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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48 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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337 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 $$...
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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 ...
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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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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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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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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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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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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 ...
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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....
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322 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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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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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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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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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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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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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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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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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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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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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 ...
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Generalizing the Cantor Set to the $n$-dimensional plane

I am interested in how to describe an $n$-dimensional cantor set. I think that it may be a good idea to develop the Cantor Set on the two-dimensional plane at first, but I am having issues figuring ...
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Push forward of the Lebesgue measure is the Haar measure of the Carnot group

I have the following problem. I have a Carnot group $(\mathbb G,*)$ which is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ is stratified as $\mathfrak g= V_1\oplus\dots\...
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The projection of density 1 point on a rectifiable set.

This post has also been posted here. Please see the comment on the linked page, useful information! Let $\Gamma\subset \mathbb R^N$ be $\mathcal H^{N-1}$-rectifiable. Then we know that $\mathcal H^{N-...
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125 views

Example of a Borel measure, which is not Borel-regular

I have asked a question to find four types of outer measures here, and I could find three of the four examples. We call an outer measure $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ Borel, if ...
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Measure of convex hulls

I'm not an expert of this kind of questions, but I can't give a satisfactory answer to the following question. Pick $x_1\dots x_n \in \mathbb{R}^m$. Is there a formula for the measure of the ...
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The $\mathcal H^{N-1}$ measure of the projection of a curve is always less then itself

Let $\Gamma \subset \mathbb R^N$ be a $\mathcal H^{N-1}$-rectifiable set such that $\mathcal H^{N-1}(\Gamma)<\infty$. Let $P(\Gamma)$ be the projection of $\Gamma$ onto the hyperplane $\{x_N=0\}$. ...
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A $\mathcal{C}^{1}$ differentiable domain and Hausdorff dimension estimates

Let us consider an open connected domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary. Suppose now, that there exists $R>0$ such that the set $\partial E \cap B_{R}(...
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Hausdorff dimension calculation related to Jarnik's theorem

Let $$F=\{x \in R:||qx||\le2q^{1-\alpha}\log q \text{ for infinitely many } q \in \mathbb{R}\}$$ Show for $\alpha>2$, $\dim_H F\le 2/\alpha$. Jarnik's theorem (By Falconer) says: Suppose $\...
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fractal curve and fractal set

Would it be correct to say that all fractal curves are fractal sets, but not all fractal sets are fractal curves? If that is correct, what would be an example of a fractal set that is not a fractal ...
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37 views

Hausdorff Measure under linear maps

We have a linear map $L : \mathbb R^n \rightarrow \mathbb R^m $ ($m\geq n$) I have two questions: How does one prove that $\mathcal H^n (L(B(x,r)))=\mathcal L^n (O^* \circ L(B(x,r)))$? (We have ...
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Inclusions for certain types of measures

Let's use the following definitions: Definition. A measure $\mu: \mathcal P(X) \to [0, \infty]$ is what some authors call a outer measure, i.e. (1) $\mu(\emptyset) = 0$. (2) If $A, A_k \...
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$\mathbf{M}$ is $\mathbf{F}$ lower semicontinuous on $\mathscr{D}_m$

I am working on a problem in Frank Morgan's Geometric Measure Theory book. What I have done so far: $T_i \to T$ under the real flat norm: $\forall \epsilon>0, \exists N$ such that $\min \{ \...
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Cone criterion for partially Lipschitz map

I'm currently reading an article in which there are some techniques that I would like to understand, I hope that someone can help me figuring out the following. Suppose we have a function $G\colon \...
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41 views

Comass of a differential form

In the wikipedia article on currents https://en.wikipedia.org/wiki/Current_%28mathematics%29 it is written that If $\omega$ is an m-form, then define its comass by $||\omega|| = \sup\{|\langle \...
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23 views

Finding the similarity dimension of a variation of the Cantor Set.

If we take the Cantor set and instead of removing the interval $[1/3, 2/3]$, we remove the open interval $[x,1-x]$, with $0<x<1/2$, will the similarity dimension change? What I think is that we ...