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Questions tagged [geometric-measure-theory]

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 such as the isoperimetric inequality and the coarea formula, and subjects such as the Plateau problem belong under this tag.

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1answer
54 views

Maximizing the value of an integral

Let $f \colon \mathbb R^N \to \mathbb R$ be a measurable, bounded function. Let $$ \mathcal A := \left\{ g \colon \mathbb R \to [0,+\infty): g \text{ is measurable and} \int_\mathbb R g =1\right\}. $$...
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1answer
28 views

Can the rank of the differential of a Lipschitz map decrease in a small neighbourhood?

Is there an example for a Lipschitz map $f:\mathbb{R}^n\to\mathbb{R}^m$ which is differentiable at $x_o$, with $\operatorname{rank} Df(x_o)=k$, such that there is no open neighbourhood $U$ of $x_0$ ...
1
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1answer
37 views

Packing Dimension as a Countable Union of Minkowski Dimension Sets

Is it true that if $X$ has packing dimension $\alpha$, then we can write $X$ as the countable union of sets $X_i$, where $X_i$ has Minkowski dimension $\alpha$. If not, which notion of dimension is ...
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1answer
19 views

What is the product of two Haar distributed unitary matrices?

I guess a product of two Haar distributed unitary matrices is also a Haar distributed unitary matrix. Is there a proof?
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1answer
56 views

How did we resolve the Banach-Taraski paradox?

I see that when Banach-Taraski paradox emerged we solved this problem by stating that not every subset is measurable so we restrict ourselves to nice sets which are measurable. But How? I'm confused a ...
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0answers
13 views

Definition of Borel vector field

My doubt is what is the definition of Borel vector field? I looked for this definition on some books of geometric measure theory and on the internet, but I didn't found. The motivation for my question ...
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2answers
31 views

Is there a generalization of the nested interval theorem in $\mathbb{R}^n$?

I'm familiar with the nested interval theorem on the real line. But is there a generalization of such a theorem in literature?
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1answer
49 views

Prove the geometric “Pigeonhole Principle”

This question is part of an introductory combinatorics class, so I don't know what measure theory is, but the question was stated as follows: Suppose $A_1, A_2,... A_k, B$ are sets which contain a '...
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1answer
19 views

Integrating the composition of a Heaviside function with a smooth function

I am trying to find how to compute an integral of the form: $\int_{R^n}{\Theta(g(x))f(x)\,dx}$, where $\Theta$ is the Heaviside function, $g(x)$ is a smooth function (a result for more general $g$ is ...
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0answers
12 views

Natural way of thinking of the definitions of the rectifiable sets and purely unrectifiable sets

I'm selfstudy Geometric Measure Theory by Frank Morgan's book and he define rectifiable sets as follows A set $E \subset \mathbb{R}^n$ is called $(\mathscr{H}^m,m)$ rectifiable if $\mathscr{H}^m(E) ...
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0answers
9 views

Integer multiplicity current without boundary is boundary of another current

Let us say we have an integer multiplicity current $T\in \mathcal{D}_n(\Omega)$, $\Omega\subset \mathbb{R}^m$, $n+1\leq m$ and $\partial T =0$. Do we always find another current $R\in \mathcal{D}_{n+1}...
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0answers
29 views

Is it true that $\mathcal H^{n-1}(\partial (A \cap B))=\mathcal H^{n-1}((\partial A) \cap B) + \mathcal H^{n-1}( A\cap ( \partial B))$?

Let $A$ and $B$ subsets of $\mathbb R ^n$, $B=B(x,r)$ an open ball, and denote the $(n-1)$-dimensional Hausdorff measure in $\mathbb R ^n$ by $\mathcal H^{n-1}$. Also assume that $\mathcal H^{n-1} (\...
6
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3answers
176 views

Compute area of a sphere through a Dirac delta

I've been having issues with integrating with a Dirac delta. To compute the area of a sphere centered at $(0,0,0)$ it seems to work just fine: $$\int_{0}^{2\pi}{\int_{0}^{\pi}{\int_{0}^{\infty}{\...
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0answers
28 views

What motivates the use of the co-area formula? What is co-area?

The motivation for the area formula is very clear. Given a set A in $\mathbf{R^n}$, and a Lipschitz function f $: \mathbf{R^n} \rightarrow \mathbf{R^{N}}$, what is the measure of the set f(A) in $R^{N}...
6
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2answers
78 views

A counter-example for integration by parts when there are “small” singularities

I am looking for a "counter-example" to integration by parts of the following type: $\Omega \subseteq \mathbb R^n$ is an open, bounded, connected domain with smooth boundary. $u,v:\bar \Omega \to \...
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0answers
21 views

Lipschitz image $\mathbf{R} \to \mathbf{R}^2$ has measure zero?

I know that of a proof of the following fact using Hausdorff measure, but is there a more elementary way to do it? Suppose $f: \mathbf{R} \to \mathbf{R}^2$ is Lipschitz. Then $\mathcal{L}^2(f(\...
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1answer
170 views

Is this function continuous/smooth?

Suppose $A \subset \mathbb{T}_2$ is a measurable set, where $\mathbb{T}_2$ is the torus group. For a fixed $n$, define the function $f_A:\mathbb{T}_2^n \rightarrow \mathbb{R}$ as $$f_A(x_1, ..., ...
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0answers
62 views

Best topics to study in order to research geometric measure theory

What are the best topics (other than GMT itself) to learn in order to pursue research in the field of geometric measure theory?
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0answers
31 views

weak* convergence of convolution between mollifiers and Radon measure

I've got a question concerning mollifiers. If $\Omega \subset \mathbb{R}^N$ is open and $\mu = (\mu_1,..., \mu_m)$ is a Radon measure in $\Omega$. Let $(\rho_{\epsilon})_{\epsilon > 0}$ be a family ...
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1answer
153 views

Request for applications of the Filling Theorem: geometric and algebraic Dehn functions are asymptotically equivalent

I am studying geometric group theory and metric spaces of bounded curvature. In particular, I was reading the seminal Gromov's Hyperbolic Groups and trying to filling the details. In the first page, ...
4
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1answer
70 views

Exact value of Hausdorff measure of two dimensional Cantor set

Let $\mathcal{C}$ denote the classical Cantor set, then it is well-known that $\mathcal{C}$ has Hausdorff dimension $\alpha = \ln 2 /\ln 3$, and its $\alpha$-dimensional Hausdorff measure is $\mathcal{...
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2answers
34 views

Borel measure determined uniquely on a base

Let $(X,\tau)$ be a topological space, $\mathcal{B}$ the $\sigma$-algebra generated by $\tau$ on $X$. Let $\mu$ be a Borel measure on $X$. Does the restriction of $\mu$ to a base for the topology $\...
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0answers
43 views

Proof of a convergence of sets in the context of Finite Perimeter sets

Let $E \subset \mathbb{R}^n$ be a set of finite perimeter that satisfies $ \mathcal{L}^n (E) < \infty$. Assume that $E$ is symmetric with respect to the hyperplane $\{x_n = 0\}$. We know that there ...
1
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1answer
50 views

formula of the Lebesgue measure of $E$ in terms of the integral regarding the Hausdorff measure

Let $E\subset\mathbb{R}^n$ be such that for the boundary of $E$ holds $\partial E=\{(1+u(x))x \mid x\in \partial B_1(0)\}$, where $u:\partial B_1(0)\to (-1,\infty)$ is a function of class $C^1$, and $\...
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0answers
24 views

Estimate/inequality releated to the Hausdorff measure

Define the diameter of a subset $Y \subseteq \mathbb R^n$ of the metric space $(\mathbb R^n, d)$ with the standard metric $d$ to be $$\operatorname{diam}(Y) := \sup_{\mathbf x,\mathbf y \in Y} d(\...
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0answers
42 views

Why is this statement “obvious” about embedded submanifolds

I've looked at many texts on rectifiable sets and I continue to see assertions that the following statement is in some sense obvious: Suppose that $N \subset \mathbf{R}^{n +k }$ is a $n$-...
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0answers
53 views

Drawing random subspaces from Grassmannian with uniform probability

Consider the Grassmannian manifold $G(M, N)$ of $M$-dimensional subspaces in $R^N$. I want to approximate (stochastically) an integral of the form $$ \int_{G(M, N)} f(v) \, dv, $$ where $f : G(M, N) \...
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0answers
12 views

Uniqueness of approximate tangent spaces with different multiplicities

Let $M \subset \mathbb R^{n+k}$ be a $\mathcal{H}^n$-measurable subset and $\theta : M \rightarrow (0,\infty)$ be $\mathcal{H}^n$ measurable such that for all $K \subset \mathbb R^{n+k}$ compact we ...
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0answers
25 views

Measure of Subspace Swept Out by Another

Sorry, it's a difficult question to give a title to, so if someone can think of a better one then feel free. I am trying to figure out how to set up a problem. Suppose we have two closed subsets $X, ...
4
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1answer
97 views

Prove that if $E \subset \mathbb{R}^n $ has finite perimeter, then almost every vertical slice has finite perimeter too.

Before explaining my problem, I recall the definitions: Let $E \subset \mathbb{R}^n$ be a Lebesgue measurable set. We say that $E$ is a set of locally finite perimeter if for every compact set $K \...
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1answer
30 views

Is the convolution by BV function a BV function?

Let $BV(\mathbb R^n)$ be the set of functions $f\in L^1(\mathbb R^n)$ such that $$\sup\Big\{\int_U f(\nabla\cdot\phi)\,dx\,|\,\phi\in C_c^1(\mathbb R^n;\mathbb R^n), |\phi|\leq 1\Big\}<\infty$$ ...
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1answer
21 views

octagon size in circle

I am using yED to draw a schematic for a sound installation. It involves a circle with a diameter of 7 metres. I need to have 8 speakers at a regular distance so I am a drawing an octagon inside the ...
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1answer
24 views

Reference request: Federer-Besicovitch structure theorem

$\newcommand{\R}{\mathbf{R}}\newcommand{\H}{\mathcal{H}}$Federer-Besicovitch prove the following result. Theorem: Let $E \subset \R^{N}$ be a purely $k$-unrectifiable set such that $E = \cup_{j=1}^...
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0answers
88 views

Application of the area formula (GMT) to derive inequality for general surface area

This question is related to Proof by induction of n-dimensional isoperimetric inequality, missing step. Discussing the same inequality but with a different approach. Gary Lawlor shows in his 2010 ...
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0answers
25 views

Any Isodiametric-type inequality for diameter and perimeter in high dimension?

This is a follow-up or part of unsolved questions from Gilles Bonnet's original post. In his original question, the cited $\textbf{isodiametric inequality}$ is about $\textit{diameter}$ and $\textit{...
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0answers
17 views

If ball can contain every set of same perimeter?

My question is related to isoperimetric inequality. We know given fixed finite perimeter, the volume of the ball (Lebesgue measure) achieves the maximum. I am wondering if the following is true, ...
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1answer
33 views

Pointwise product of Borel sets is Borel?

If $A \subseteq \mathbb{R}$ and $B \subseteq \mathbb{R}^2$ are Borel, is $AB = \{ab:a\in A, b \in B\}$ always a Borel set? Scalar multiplication is continuous but not Lipschitz or injective, so its ...
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1answer
25 views

Are there simple criteria for determining if there is a finite, non-zero Hausdorff measure of a set?

It’s kind of a wide question, so I’d like to motivate it: When trying to determine the Hausdorff dimension of a set, I found that it is often reasonably easy to find it heuristically, but very ...
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1answer
47 views

understanding an estimation of the perimeter of sets, $|P(\hat{V})-P(V)|\le P(B_r(x))$

Let $V$ be a Borel set in $\mathbb{R}^n$ such that the Lebesgue measure of $V$, $|V|$, satisfies $|V|\approx |B_1(0)|$, but $|V|\neq |B_1(0)|$ (i.e. $|V|$ is slightly greater or less than $|B_1(0)|$). ...
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1answer
93 views

Lebesgue measure of intersection of sets

Let $A,B\subset\mathbb{R}^n$ two Borel sets with finite and positive Lebesgue measure such that $|A|=|B|$, where $|A|$ denotes the Lebesgue measure of $A$, and such that $A\triangle B \subset\subset C(...
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1answer
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perimeter of sets. How do $P(E,\Omega)$ and $P(E\cap \Omega,\mathbb{R}^n)$ relate?

Let $\Omega$ be an open set in $\mathbb{R}^n$ and $E$ a Borel set. The relative perimeter of $E$ w.r.t. $\Omega$ is defined as $$P(E,\Omega)=\sup\left\{\int_{\Omega}\chi_E(x) \mathrm{div}\boldsymbol{\...
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0answers
31 views

Isoperimetric Inequality via Brunn-Minkowski is valid for every norm?

Suppose I have got $A\subseteq\mathbb{R}^n$ such that $|A|<+\infty$, where $|\cdot| $ indicates the Lebesgue measure. I define $$|\partial A|:=\liminf_{\epsilon\to0}\frac{|(A+\epsilon K)\backslash ...
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2answers
46 views

A geometric probability question

Find the probability of distance of two points ,which are selected in $[0,a]$ closed interval, is less than $ka$ $k \lt 1$ What did I write : $P(A)$ = (Area measure of set $A$)/(Area measure of set $...
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2answers
133 views

When Brunn-Minkowski inequatily $(m(A+B))^{1/d} \geq (m(A))^{1/d} + (m(B))^{1/d}$ becomes equality?

Let $A$ and $B$ be two non-empty compact subsets of $\mathbb{R}^d$. Brunn-Minkowski inequality gives $(m(A+B))^{1/d} \geq (m(A))^{1/d} + (m(B))^{1/d}$. But how to prove the following? $(m(A+B))^{...
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1answer
44 views

Is there a set whose essential boundary is the Cantor set?

I would like to construct a set $E\subset [0,1]$ whose essential boundary is the Cantor set $C$. More precisely, let us denote by $d_E(x)$ the limit $\lim_{r\to 0+} \frac{\lambda(E\cap (x-r, x+r))}{...
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1answer
43 views

relation between the Lebesgue measure and the perimeter of a set

Let $(E_j)_j\subset \mathbb{R}^n$ be a sequence of Borel sets with $P(E_j)<\infty$ for all $j$, where $P(E)=\sup\left\{\int_{\mathbb{R}^n}\chi_E(x) \mathrm{div}\boldsymbol{\phi}(x) \, \mathrm{d}x : ...
3
votes
1answer
55 views

Doe a smooth function map positive measure sets to positive measure sets

Suppose $f: X \subset R^n \to R^n$ is a smooth function (for example $C^2$ function), and for each $y \in R^n$, the set $f^{-1}(y)$ is finite. Do we have $f(A)$ is a positive measure set if $A$ has ...
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0answers
82 views

Measure-theoretic boundary of a fat Cantor set

Let $C_\lambda\subset [0,1]$ be the fat Cantor set of parameter $\lambda$ (which is constructed as the usual Cantor set, removing at the $n$-th step the middle intervals of length $\lambda / 3^n$). ...
3
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1answer
70 views

Hausdorff measure on non separable spaces

In his book Geometry of Sets and Measures in Euclidean Spaces, Pertti Mattila defines the Hausdorff measures via the Carathéodory's construction (chap.4). My doubt ...
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0answers
51 views

Capacity and Hausdorff measure

My question arise from Thm 4.16 page 179 of the book by Evans & Gariepy, Measure Theory and Fine Properties of Functions (revised edition). I want to prove the following: If $\mathcal{H}^{n-p}(A) ...