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 ...

learn more… | top users | synonyms

3
votes
0answers
139 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 ...
3
votes
0answers
130 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) \; ...
3
votes
0answers
172 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 ...
2
votes
3answers
337 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 ...
2
votes
3answers
119 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 ...
2
votes
2answers
217 views

Sub-dimensional linear subspaces of $\mathbb{R}^{n}$ have measure zero.

I would appreciate it if someone could refer me to a proof (or simply give one here) for the statement in the title. That is: If $k<n$, then every $k-$dimensional subspace of $\mathbb{R}^{n}$ has ...
2
votes
1answer
291 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
101 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 ...
2
votes
1answer
52 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 $$ F:\mathbb{R}^m\to [0,\infty]\\y\mapsto H^{s}(K\cap f^{-1}(\{y\})) $$ ...
2
votes
1answer
45 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 ...
2
votes
1answer
59 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 ...
2
votes
1answer
134 views

Two Definitions of Minkowski Dimension

I'm currently reading a paper. Let $F\subset\mathbb R^n$ and $\epsilon\gt0$, the paper defined $m^s(F):=\liminf_{\epsilon \to 0}\epsilon^{s-n}\lambda(F_\epsilon)$ and $M^s(F):=\limsup_{\epsilon \to ...
2
votes
1answer
159 views

A question about the proof of Rademacher theorem

I'm referring to the proof of Rademacher theorem due to C.B.Morrey (i'm reading it on Simon: 'Lectures on geometric measure theory').\ The proof can be summarized in the following steps:\ 1)For every ...
2
votes
2answers
60 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 ...
2
votes
2answers
89 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 ...
2
votes
1answer
30 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 ...
2
votes
1answer
102 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) ...
2
votes
1answer
87 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 ...
2
votes
1answer
46 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 ...
2
votes
1answer
158 views

About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given ...
2
votes
1answer
217 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
192 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
1answer
142 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 ...
2
votes
0answers
61 views

Basic Fourier analysis explanation needed wrt a function $f$ and a finite Borel measure $\mu$

An extract from Chapter 12 of Matilla's Geometry of Sets and Measure on Euclidean Spaces I do not believe that formulas (12.1-12.3) are easily seen to be valid. I do not understand what ...
2
votes
0answers
28 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} ...
2
votes
0answers
37 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 ...
2
votes
0answers
26 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 ...
2
votes
0answers
47 views

Problem with definition of pushforward density

Let $X$, $Y$ be smooth manifolds and let $\pi \colon X \to Y$ be a submersion. Then for every $y \in Y$ the set $W_y = \pi^{-1}(y)$ is a submanifold in $X$. Let $\mu \in \Gamma(|\Lambda| X)$ be a ...
2
votes
1answer
72 views

Upper Bound on Surface Area of Algebraic Surfaces

I have a family of multi-variate polynomials, with bounded degree and a bound on the norm of the coefficients, say for example, all multi variate polynomials of the form: $\{ a_1 xy + a_2x + a_3xyz: ...
2
votes
0answers
51 views

$L^1$ approximate continuity points of epigraph of a continuous function

Consider a continuous function $f\colon \mathbb R\to \mathbb R$. Let $F(x,y):=\mathrm{sign\,}(y-f(x))$. Is it true that for a.e. $x\in \mathbb R$ the point $(x,f(x))$ is not a point of $L^1$ ...
2
votes
0answers
62 views

Measures whose projections are absolutely continuous

Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt rotations) projections of $\mu$ to lines on the plane are absolutely ...
2
votes
1answer
106 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, ...
1
vote
2answers
97 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 ...
1
vote
1answer
86 views

Is Hausdorff measure continuous with respect to Hausdorff distance?

Say I have a sequence of surfaces, $S_n$ in $\mathbb{R}^k$ with $d$-dimensional Hausdorff measure $m_d(S_n)$. Say $S_n$ converges in the Hausdorff distance sense to a surface $S$ with $d$-dimensional ...
1
vote
2answers
38 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 ...
1
vote
2answers
34 views

A conjecture about lines and points in the plane

Let $ l_1, l_2, \ldots$ be an infinite sequence of lines in the plane, and let $(a_1, b_1), (a_2, b_2), \ldots$ be an infinite sequence of pairs of points such that $a_i, b_i \in l_i$ and $a_i \neq ...
1
vote
1answer
27 views

Show that an irregular 1-set in the plane is totally disconnected

A $1$-set is a Borel set such that $0 < \mathcal{H}^1(A) < \infty$, where $\mathcal{H}^s$ is the Hausdorff measure. Let $A$ be an irregular $1$-set in the plane. Deduce from the theorem below ...
1
vote
1answer
49 views

A question on Hausdorff measure

This question is part of a homework assignment. Considering the hausdorff measure $\mathcal{H}_{2}$ on $\mathbb{R}^{3}$, I need to compute the measure of the unit cube: $A = \{(x,y,z) \in ...
1
vote
1answer
46 views

Does the squared root $\sqrt{|\cdot|}$ belong to $BV((-1,1))$?

Does the squared root belong $\sqrt{|\cdot|}$ to $BV((-1,1))$? In an affirmative case, what is its derivative in the distributional sense, i.e., what is the Radon measure $\mu$ such that $\mu=Du$ ?
1
vote
1answer
112 views

The problem of concentration (Clarification of statement in Evans: Weak Convergence Methods for Nonlinear PDE)

I am working through Evan's book on Weak Convergence Methods for Nonlinear PDE. He assumes that $U$ is an open bounded smooth subset of $\mathbb{R}^n$ and that $1<q<n$. In ($\S$D) concerning ...
1
vote
1answer
211 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 ...
1
vote
1answer
25 views

Fractal Dimension of $C_{\frac{1}{3}}\times[0,1]$

I wonder what is the dimension of the fractal set given by the product of the unit interval $[0,1]$ by the thirds-cantor-set ($C_\frac{1}{3}=\bigcap_n C_n$ where $C_0=[0,1],C_1=[0,\frac 1 3]\cup[\frac ...
1
vote
1answer
43 views

Is this c the same as that c?

Are the highlighted $c$'s the same or should it be $c_1$ and $c_2$.
1
vote
1answer
29 views

Proving that the Bernoulli self similar measure is doubling

Let $\mu_p$ a measure which is the push forward of the bernouli product measure $(p,1-p)^\mathbb N$. Let S=$\{f_1,\dots f_m\}$ an IFS, a system of functions with attractor $K$, means ...
1
vote
1answer
36 views

Use of Banach-like Covering theorem

where I cannot see how the highlighted equation has been obtained. I cannot understand how the setminus operation has been justified.
1
vote
1answer
36 views

Proving that a mass distribution has positive Lebesgue measure

I am confused in this proof about how we obtain $\int f(u) \, d\mu(u) = \int f(u)g(u) \, d\mu(u)$ and how Plancherels theorem has been applied in $(6.6)$. Furthermore, I cannot understand how if ...
1
vote
1answer
39 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 ...
1
vote
1answer
33 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. ...
1
vote
1answer
42 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 ...
1
vote
1answer
29 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 ...