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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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7 views

A covering argument for metric Jacobian

Given a Lipschitz map between Carnot Groups $ f : G_1 \to G_2$, with homogeneous dilations $ \delta^1_s, \delta^2_s$, we have the almost everywhere Pansu derivative $ D_H f(x)(y) = \lim_{s\to 0} ...
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1answer
25 views

Estimating the missing points of a 3D point cloud

Consider a cloud of N points (forming a smooth 3D object), in which n points are missing. Also, consider that there is no prior knowledge about the original shape of the point cloud. The only ...
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10 views

Showing that a precursor to the packing measure is not a measure

I am trying to prove the highlighted sentence. What countable dense sets should I consider? and how am I trying to prove this is not a measure?
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1answer
16 views

Showing the equivalence of different definitions of the box-counting dimension

I am trying to prove the statment "by taking logarithms...". $\lim\limits_{\delta \rightarrow 0} \frac{log(N_{4\delta}(F))}{log(\frac{1}{4 \delta})} \leq \lim\limits_{\delta \rightarrow 0} ...
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1answer
67 views

Evaluating the limit for the Minkowski content of $F_{\alpha}=\{0,1,\frac{1}{2^{\alpha}},\frac{1}{3^{\alpha}},\frac{1}{4^{\alpha}},\dots\}$.

The Minkowski content is defined as $\displaystyle M_{\beta}(A)=\lim_{\delta \rightarrow 0} \frac{\mu(A_{\delta})}{(2\delta)^{1-\beta}}$ where $0 < \beta < 1$, $A \subset \mathbb{R}$, and ...
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39 views

Standard reference for “coarea formula”?

Is there a standard analysis reference for the first formula given in these lecture notes: $$\int_D h(x)\,dx = \int_{-\infty}^\infty dt\int_{D_f(t)}\frac{h(y)}{|\nabla f(y)}\,dS_y$$ where $dS_y$ is ...
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1answer
26 views

question on existence of open set

Let $U$ be a bounded open set in $\mathbb{R}^n$ and $A$ be an open subset of $U$. Fixed $\epsilon >0$. Does there exist an open set $B \subset U$ such that $B \cap \overline{U} \ne \emptyset$ and ...
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1answer
29 views

Hausdorff dimension of a Modified Cantor like set

Suppose you have the unit interval $[0,1]$. For the first iteration you remove the segment $(1/5,3/5)$. So you are left with two intervals of lengths $1/5$ and $2/5$. You now repeat the process on the ...
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30 views

Show the Hausdorff dimension of a set of numbers with digit 5

I can show heuristically that the answer is $log(9)/log(10)$ but I am struggling to prove this rigorously. This is using the construction that after the first iteration we have $9$ intervals, length ...
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1answer
15 views

Bi-Lipschitz invariance of the box-counting dimension.

I would like to prove that the box counting dimension is invariant under a bi-Lipschitz transformation. We have that $f$ is bi-Lipschitz if there exists $c_1, c_2$ such that $0 < c_1 \leq c_2 < ...
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1answer
26 views

Relationship between the Hausdorff dimension and the Box-counting dimension

In Fractal Geometry by Falconer the author writes: If $1<\mathcal H^s(F)=\lim_{\delta\to0}\mathcal H_\delta^s(F)$ then $\log N_\delta(F)+s\log\delta>0$ if $\delta$ is sufficiently small. ...
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1answer
31 views

Problem in the proof of the dimension of the Cantor set

From the proof of the Hausdorff dimension of the middle third Cantor set. I cannot understand the last sentence in this proof. I cannot see how we have counted $2^j \leq \sum_i 2^j3^s|U_i|^s$
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1answer
35 views

Does interval spacing effect Hausdorff dimension of Cantor set?

Let $C=\bigcap_{j=0}^{2^n}C_j$, $C_0=[0,1]$, and the intervals in the construction of each stage of $C_j$ consists of removing the center 1/3 from the $j-1$ stage intervals. In other words, the ...
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1answer
37 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 ...
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1answer
48 views

Prove that this function is Borel measurable

Prove that if $s\ge 0$, $f:\mathbb{R}^n\to\mathbb{R}^m$ is continuous and $K\subset\mathbb{R}^n$ is compact, then the function $$ F:\mathbb{R}^m\to [0,\infty]\\y\mapsto H^{s}(K\cap f^{-1}(\{y\})) $$ ...
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12 views

$ H^{n-1} (spt \mu _E - \partial ^{*}E)=0 ?$

In Federer's Theorem, $ H^{n-1} (\partial ^{m}E - \partial ^{*}E)=0 $, where $E$ is a set of finite perimeter in $ \mathbb R^n $, $\partial ^{m}E$ is the measure theoretical boundary of E, and ...
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1answer
26 views

Measure defined in an atypical way

I was reading a paper when I found this ($\partial \Omega$ refers to the boundary of $\Omega$ and $\nabla$ to the gradient operator,$\nabla f = (\partial_{i}f)_{i} $ ). Let $\Omega \subset ...
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1answer
23 views

Two Radon measures and mutual singularity

Let $\mu$ and $\lambda$ be Radon measures on $\mathbb{R^n}$. Show that $\mu$ and $\lambda$ are mutually singular iff $D(\mu,\lambda,x)=\infty$ for $\mu$ almost all $x \in \mathbb{R^n}$. I have looked ...
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1answer
34 views

Measure derivatives and the chain rule

Let $\mu$ and $\lambda$ be Radon measures on $\mathbb{R^n}$ such that $\mu << \lambda$. Prove that $\displaystyle \int D(\mu,\lambda,x)^2 d\lambda x= \int D(\mu,\lambda,x)d\mu x$. Is it ...
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23 views

Density and $\lambda$ - measurability of a Radon measure

Let $\lambda$ be a Radon measure on $\mathbb{R^n}$ and $A \subset \mathbb{R^n}$. Show that $\displaystyle \lim_{r \rightarrow 0} \frac{\lambda(A \cap B(x,r))}{\lambda(B(x,r))}=0 \ \ \ \text{for ...
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1answer
23 views

Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.

Let $\Omega\subset\mathbb{R}^N$ be a open, unbounded and connected set ($N\ge 2$). Let $m$ and $\mathcal{H}^{N-1}$ denote respectively, Lebesgue and $(N-1)$-Hausdorff measures. Suppose that ...
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1answer
51 views

Choice of number in the proof the 5-r covering theorem

Why has the number 3 been chosen? I have tried drawing this and it seems wrong (its not). The balls definitely dont seem to be disjoint either. It would seem that if a particular $x$ has $r(x)$ ...
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1answer
24 views

Showing that the upper packing dimension is the packing dimension

I cannot see how the first inclusion in this proof works. $P$ is the maximum number of disjoint $B(\epsilon/2)$ with centres in $A$ and the following will help. Moreover I cannot see how it ...
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1answer
20 views

Equalities for the Upper and Lower Minkowski dimension definition

In a Geometric Measure Theory textbook the following was written: I cannot see how any of these equalities hold and dont believe they are obvious. If they are relatively obvious could someone ...
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1answer
23 views

Properties of the Hausdorff measure

This comes from a book on geometric measure theory in a chapter introducing the Hausdorff measure $\mathcal{H^t}$. I cannot see in this proof how $\sum_i d(E_i)^s \leq \mathcal{H^s_{\delta}}(A)+1$ ...
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10 views

existence of symmetrization algorithm

For symmetrization of a Borel set, the construction of an an explicit algorithm (i.e. a sequence of symmetrization steps that will lead to a ball) is an open question. But have we proved the ...
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1answer
29 views

Symmetrization Methods

I was wondering if I could get a list of the symmetrization methods out there i.e. methods that rigidly transform a set A into it's equimeasure ball $A^{*}$. Here are some: a) Steiner Symmetrization ...
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194 views

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
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29 views

How to get between the equivalent definitions of Newtonian capacity?

Here are the two definitions: (1) $$\mathrm{Cap}(A)=\left[\inf \left\{\int\int |x-y|^{d-2}\mu(dx)\mu(dy):\mu \; \text{a probability measure on} \; A \right\}\right]^{-1}$$ and (2) ...
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1answer
38 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, ...
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0answers
35 views

Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets. For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...
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1answer
53 views

What is the minimum Number of closed balls covering a boundary as radius $r\to 0$?

Here is the problem: Given compact set $A\subset \mathbb{R}^{d}$, cover $\partial A$ by closed balls $\{B_{i,\varepsilon}\}_{i=1}^{n}$ , with minimum overlap. Can we express n as a factor of ...
2
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1answer
41 views

References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
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2answers
124 views

Best textbook for Geometric Measure Theory

I was wondering what is the best textbook for Geometric Measure Theory for self study. I am looking for one that isnt excessively detailed or long either as I found Rana's Introduction to measure ...
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28 views

Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function ...
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1answer
29 views

Measure on Locally Compact, Separable metric space

Simon (Geometric Measure Theory) says: If $X$ is a locally compact, separable metric space and $\mu(K) < \infty$ for all $K$ compact, then $X=\cup_{i=1}^\infty U_i$ where $U_i$ are open and ...
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31 views

Computing equilibrium measure for Borel sets eg. Ball

I am asking for methods to compute equilibrium measures. The more the better. Here is the definition of equilibrium measure in the Brownian motion setting: Let $\gamma=\sup\{t\in [0,T]: B_{t}\in ...
2
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1answer
55 views

Constructing an example related to Vitali's covering theorem

An exercise in Fremlin's measure theory vol 2 asks to construct a family $\mathcal I$ of open intervals in the real line such that every point of $\mathbb{R}$ belongs to arbitrarily small intervals of ...
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0answers
31 views

Difference of two subharmonic functions and signed measures

One of the reasons subharmonic functions are interesting is that if you take their laplacian, you get a measure (and conversely any finite Radon measure with compact support can be obtained in this ...
3
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1answer
88 views

The walk of a knife

"A knife is slowly moved parallel to itself over the top of a cake. At each instant the knife is poised so that it could cut a unique slice of the cake. As time goes by the potential slice increases ...
2
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1answer
100 views

Lipschitz manifold and semi-algebraic is Lipschitz graph?

It is known that there are Lipschitz manifolds that are not Lipschitz graphs and $C^1$ manifold is locally $C^1$ graph. If $M\subset \mathbb{R}^m$ is a Lipschitz manifold (with the outer distance) ...
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1answer
82 views

Measurability of upper and lower derivatives of Radon measures

Let $\mu$ and $\nu$ be Radon measures in $\mathbb R^N$. Define their upper and lower derivatives by $$ \overline{D}_\nu\mu(x):=\limsup_{r\to0}\frac{\mu(B_r(x))}{\nu(B_r(x))},\qquad ...
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2answers
53 views

Graph (or manifold) Lipschitz satisfy the sphere (ball) condition?

Consider $\varphi: U\subset \mathbb{R}^{n-1}\to \mathbb{R}$ a Lipschitz function and $\Omega=Graph(\varphi)$, i.e., $$\Omega=\{x=(x_1,...,x_n)\in U\times\mathbb{R};x_n=\varphi(x_1,...,x_{n-1})\}.$$ ...
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1answer
34 views

Projection of measurable sets

If $ X $ and $ Y $ are metric spaces, $ f: X \rightarrow Y $ is lipschitzian and $ H^k $ is the Haussdorf measure, it is easy to check that $ f(A) $ is $H^k $-measurable whenever $ A $ is $H^k ...
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2answers
115 views

Borel sets and measurability

Is it always possible to construct a measure $ \mu $ on a Hausdorff space Y such that the $ \mu $-measurable sets are exactly the Borel sets of Y? By Theorem in 2.2.13 of Federer's book this question ...
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1answer
29 views

Hausdorff dimension and accumulation points on a smooth curve

I am wondering about the following, possibly naive, question. Suppose I have a smooth curve, which intersects the horizontal axis in a manner that leads to an accumulation point. More precisely, ...
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37 views

Borel outer measure and Borel measure

I have these two definitions: Given $\ (X,\mathcal{M},\mu)$ measure space, the measure $\ \mu$ is Borel measure if $\ \mathcal{M}=\mathcal{B}(X) $ Given $\ \phi:X\to[-\infty,+\infty]$ be an outer ...
3
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0answers
77 views

Lebesgue density theorem for compact metric spaces.

Let $X$ be a compact metric space (with balls $B_{\varepsilon }(x)$), $\mu $ a Borel probability measure, and $A$ a Borel set with positive probability. Do we have that $\lim_{\varepsilon ...
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0answers
25 views

Show that $d\log f$ is a 1-current on a 1-dimensional complex manifold

I am having trouble with this problem (and it might be because I have the formulation slightly off). I need to show that $d\log f$ is a 1-current on a 1-dimensional complex manifold $M$. This means ...
1
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1answer
67 views

A Borel measure defines semi-continuous function?

Let $X$ be a metric space with outer measure $\mu$, which is assumed to be a Borel measure, i.e., all Borel sets are measurable. For a fixed subset $A\subset X$ (not necessarily measurable, but you ...