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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 ...

4
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1answer
57 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
32 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 $\...
1
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0answers
31 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
vote
1answer
41 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
8 views

Minkowski content and the invariant measure on $\mathrm{SL}_2(\mathbb R)$?

Denote the Minkowski content measure on $\mathrm{SL}_2(\mathbb R)$ as $\mathrm d\sigma$ (think of as a subset of $\mathrm{GL}_2(\mathbb R)$), and the invariant measure (under matrix multiplication) on ...
0
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0answers
21 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
39 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
24 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) \...
0
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0answers
8 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 ...
2
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0answers
23 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
88 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 \...
1
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1answer
27 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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votes
1answer
19 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 ...
1
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1answer
21 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}^...
2
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0answers
77 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
20 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
16 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, ...
1
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1answer
29 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 ...
1
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1answer
24 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 ...
0
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1answer
41 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 $|B_1(0)|$). ...
2
votes
1answer
69 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(...
3
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1answer
38 views

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
30 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 ...
0
votes
2answers
44 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 $...
3
votes
2answers
118 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))^{...
2
votes
1answer
35 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))}{...
1
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1answer
39 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
53 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 ...
0
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0answers
49 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
votes
1answer
69 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
38 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) ...
1
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0answers
48 views

Is finiteness of Assouad dimension a topological invariant for compact metric spaces?

A metric space is called doubling if there is some $C>0$ such that for any $r>0$ any ball of radius $r$ can be covered by $C$ balls of radius $r/2$. This is equivalent to having finite so-...
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0answers
19 views

Q: The relationship between convergence of sequence (fn) almost everywhere, almost uniformly and convergence in measure

Q:how we can discribe relationship between convergence of sequence (fn) almost everywhere, almost uniformly and convergence in measure?
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0answers
19 views

How can I prove that an outer Carathéodory measure is lower semicontinuous?

The problem is the following: Let $\varphi$ be an outer Carathéodory measure on $\mathbb{R}$ and let $f(x):= \varphi(I_x)$ where $I_x$ is an open interval of fixed length centered at $x$. Prove that ...
7
votes
1answer
132 views

Constructing a function with constant line integral in $\mathbb{R}^n$?

Suppose, $\Omega \subset \mathbb{R}^n$ is a bounded convex set. If, there is an integrable function $f:\Omega \to \mathbb{R}$ s.t. $$\int_{\Omega \cap \ell} f = 1$$ for every line $\ell \subset \...
1
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3answers
82 views

Minimizing total variation with small norm in a compact set

Take $\Omega\subseteq\mathbb R^n$ to be compact. Suppose for $t>0$ we define $f$ via the variational problem $$ \begin{array}{rl} \inf_{f\in\mathrm{BV}(\mathbb R^n)} & \mathrm{TV}[f]\\ \textrm{...
1
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1answer
29 views

Convergence to zero in total variation: Seeking a contradiction

Suppose I have a compact measurable set $\Omega\subseteq\mathbb R^n$ and a sequence of functions $\{f_k\}_{k=1}^\infty\subset C^\infty(\mathbb R^n)$ with the following set of properties: $f_k\geq0$ $\...
2
votes
1answer
28 views

The closure of an open subset in $\mathbb{R}^d$ is Ahlfors regular?

I have a question about Ahlfors regular space. Let $U$ be a bounded open subset in $\mathbb{R}^d$. We denote by $m$ the Lebesgue measure on $U$. Then, can we show the following? There exists a ...
3
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1answer
121 views

Why is 3 a bad constant in the Vitali covering lemma (infinite version)?

The Hardy-Littlewood maximal function inequality can be proved using the Vitali covering lemma (infinte version) with constant k=5. Actually it can be shown that the constant in the lemma just needs ...
2
votes
1answer
71 views

Measure of set where functions are simultaneously bounded by their respective averages

This pertains to an exercise in Maggi's "Sets of Finite Perimeter and Geometric Variational Problems"-- specifically Exercise 1.14: If $u_k$, $1\leq k\leq N$ are summable on $\mathbb R^n$ with ...
1
vote
1answer
115 views

Set of Hausdorff dimension $\alpha$

I would like to construct a set $\Sigma\subset\mathbb{R}^n$ such that $$ mr^\alpha\leq\mathscr{H}^\alpha(B(0,r)\cap\Sigma)\leq M r^\alpha, $$ for all $0\leq r\leq1$ and some $0<m\leq M<\infty$. ...
3
votes
1answer
63 views

Schwarz symmetrization is equimeasurable

Suppose $\Omega\subset\mathbb{R^2}$ is open and bounded, and let $f:\Omega\rightarrow [0,\infty)$ be measurable. Moreover, let $\Omega^{\ast}$ denote the closed disk with midpoint $0\in\mathbb{R}^2$ ...
1
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0answers
28 views

Is it possible to define an integration measure on a presymplectic manifold induced by the presymplectic structure?

Let us consider a symplectic manifold $(M,\omega)$. By means of this structure, I can easily define a symplectic measure $\mu(M)$ on $M$ by considering the "right number" of wedge product of the ...
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votes
0answers
22 views

Change of variables between quadrilaterals - Rayleigh quotient

A - Vertex at bottom left B - Vertex at bottom right K - Vertex at top left of blue quadrilateral C - vertex at top left of brown quadrilateral L - vertex at top right of blue quadrilateral F - ...
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0answers
170 views

Minimum area contained between measurable set and translate by $\lambda$: A strengthening of 2018 USA TSTST #9

Question Given $\lambda\in\mathbb{R}^+$, what is the smallest possible $c$ for which, given any measurable region $\mathcal{P}$ in the plane with measure $1$, there always exists a vector $\mathbf{v}$...
2
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0answers
36 views

Mass of current with different Norms

I've been reading the chapter about currents in Simon's Geometric Measure Theory (see e.g. http://web.stanford.edu/class/math285/ts-gmt.pdf). In section 2 he defines the Mass of an $n$-current $T$ by $...
2
votes
1answer
131 views

Domains for which the divergence theorem holds

In the book Elliptic partial differential equations of second order written by Gilbarg and Trudinger, I saw the following sentence on page 17 in section 2.4 Green’s Representation: As a prelude to ...
2
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0answers
37 views

Question about smoothing differential forms/currents

My personal motivation: Let $\omega$ be a $C^1$ closed differential $k$-form on a smooth manifold $M$. Then there exists a $C^\infty$ closed differential $k$-form $\alpha$ and a $C^1$ exact form $d\...
0
votes
0answers
21 views

Number of positive Lebesgue measurable sets. [duplicate]

Does there exist uncountably many Lebesgue measurable subsets $\{E_i\}$ in $\mathbb{R}$ with $\lambda(E_i)>0$ and $E_i\cap E_j=\emptyset$.
3
votes
1answer
121 views

Most valuable rectangle contained in an equilateral triangle

A measure $V$ (for "value") is defined on an equilateral triangle. $V$ is absolutely-continuous with respect to the Lebesgue measure, and the value of the entire triangle is $1$. What is the largest $...