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 geometry, Riemannian geomerty, sub-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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0answers
30 views

Can objects with different numbers of dimensions be compared?

To elaborate on the title: In many places, I've seen people claiming that objects residing in different dimensional spaces are not commensurable; that is, there is no way to state whether a cube of ...
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23 views

Usefulness of Lebesgue points

Let $(\mathcal{X},d)$ be a metric space, $\mu$ a locally finite Borel measure of $(\mathcal{X},d)$ and $f$ be a real locally $\mu$-integrable function of $(\mathcal{X},d)$. We say that $x \in \mathcal{...
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If $R\subseteq\Bbb{R}^n$ is a rectangle then $m(R)=0\Leftrightarrow v(R)=0\Leftrightarrow\overset{°}R=\varnothing$.

What shown below is a reference from "Analysis on manifolds" by James R. Munkres Definition Let $A$ a subset of $\Bbb{R}^n$. We say $A$ has measure zero in $\Bbb{R}^n$ if for every $\epsilon &...
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24 views

Can positivity of currents implies positivity of forms?

Let $\alpha$ and $\beta$ be 2 continuous (or smooth) forms of $(1,1)$-type on a complex manifold $X$. Of course they can be considered as currents. Assume $\alpha\geq \beta$ in the sense of currents. ...
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104 views

Inequality of Hausdorff measures for convex sets $\mathfrak{H}^{n-1}(\partial E)\le \mathfrak{H}^{n-1}(\partial F)$

I'm preparing for an exam on the calculus of variations and I need help in solving this exercise from an old exam text (actually it's only a part of a bigger exercise but its parts are quite ...
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1answer
322 views
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Help with the proof that $E\subset \mathbb{R}$ with finite perimeter and area has to be equal to the finite union of bounded intervals

I'm preparing for an exams and I've problems solving this exercise from an old exam, any help will be welcomed Let $E\subset \mathbb{R}$ be a measurable set with $\mathfrak{L}^1(E)<\infty$ and $$P(...
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1answer
27 views

Convergence of set vs convergence of perimeter

Let $\Omega\subseteq\mathbb{R}^n$ be an open set and $E$ be a set of finite perimeter in $\Omega$ (i.e., the indicator function $\chi_E\in BV(\Omega)$), $\|\partial E \|(\Omega)$ the perimeter of $E$ ...
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2answers
153 views

The volume of the image of a map with vanishing Jacobian is zero

Let $\Omega \subseteq \mathbb{R}^n$ be a nice domain with smooth boundary (say a ball), and let $f:\Omega \to \mathbb{R}^n$ be smooth. Set $\Omega_0=\{ x \in \Omega \, | \, \det df_x =0 \} $ Is ...
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Integral equality implies Hausdorff measure equality?

Suppose that the following equality holds true: $$\int_{\mathbb{R}}\mathcal{H}^{n-1}\left(B \cap A_r \right)dr=\int_{\mathbb{R}}\mathcal{H}^{n-1}\left(B \cap C_r \right)dr, \quad \forall \, B \in \...
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39 views

Isoperimetric inequality for bounded domains in $\mathbb{R}^n$

I'm looking for the proof of the following/reference to such proof. At the end of the day, my goal is to confirm if this inequality holds. Let $H_{n-1}$ be the $n-1$ dimensional Hausdorff measure ...
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39 views

Beyond Radon-Nykodim: Besicovitch theorem

I found, with no reference, the following theorem which is called Besicovitch derivation theorem. Do you know any article/book where I can find this powerful result? Let $\Omega\subset \mathbb R^n$ ...
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Let $𝑓:\mathbb{R}→[0,\infty]$ whit $t→ \mathcal{H}^{n-2}(M \cap \mu^{-1}\{t\})$, Show that $f$ is measurable.

My Attemp: Let $\mu:\mathbb{R}^n→ \mathbb{R}$ is a lipschitz function, and M is a locally $\mathcal{H}^{n-1}$-rectifiable set in $\mathbb{R}^n$. I know I take the restriction $f|_{t}$, and I consider $...
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1answer
34 views

Simple question about Lebesgue density points of open sets

I have a question regarding Lebesgue density points. Assume that $E \subset \Omega$, where $E$ and $\Omega$ are two open subset of $\mathbb{R}^n$; now define the function $$D(x):=\lim_{r \to 0^+} \...
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1answer
57 views

A question on the relation of two different forms of the Spectral Theorem for bounded operators

I am going through some spectral theory, and I have found two results under this name. I state this results: (I) Spectral Theorem I: Let $\mathcal{H}$ be a separable Hilbert space. If $A\in L(\...
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An explicit formula for conditional expectations via differentiation theorem

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $Z:\Omega\to[0,\infty)$ be a bounded random variable. Let $(\mathcal{W},d)$ be a metric space and $W:\Omega \to \mathcal{W}$ be a ...
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1answer
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Measure of error in smoothness of approximation of sphere

I'm meshing a sphere and am solving a physics problem on this. What I want to show is that the error in the model scales like$$ \varepsilon = \epsilon^p, $$ where $\epsilon$ is the "error" in the ...
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Check consistency and extend, if possible, measures of union and intersection of four sets of quantum-information-theoretic interest

I have four sets--let us denote them $A_i, i=1,\ldots,4$--of quantum-information-theoretic interest (https://arxiv.org/abs/2004.06745, https://arxiv.org/abs/1905.09228). The set $A_1$ contains the ...
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Finite set of vectors approximating a unit ball.

I am having difficulty proving that a unit ball can be approximated with a set of finite vectors. Specifically, I want to bound the error of the following approximation. Let $D$ be a uniform ...
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25 views

Metric to measure “jigsaw fitting”

I am looking for a metric that would help cluster 2D lines (or 3D shapes). For example, in this Figure, the black and red lines should belong to one cluster, while the blue one should belong to ...
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1answer
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Subadditivity of Lebesgue inner measure

Im in a trouble. I want to prove this proposition: Let $A,B \subseteq \Re^n $ such that $d(A,B)>0$. Then, $ m_*(A\cup B) \le m_*(A) + m_*(B) $ . Where $m_*$ is the Lebesgue inner measure. I don'...
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1answer
58 views

Why does the length of a string remains same inspite of deformation?

I got stuck thinking about a question which seems to be very trivial. The question is that if we consider a string(basically a line segement) and now if we deform the string to change its geometry ...
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Motivation of geometric analysis

Geometric analysis in some sense tries to make sense of calculus on spaces where we don't require any smoothness (for example metric measure spaces). What are the type of problems that motivate these ...
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1answer
29 views

Does this distance measure have a name?

I'm looking for discussion / literature recommendations of a (possibly slightly weird) measure function: perhaps a weighted version of taxicab distance? This is not really my area and so I'd be super ...
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1answer
35 views

Is there a mapping of the disk with Jacobian greater than 1 that respect the boundary?

This is a follow-up of this question. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Does there exist a smooth map $f:D \to D$ such that $\det df >1$ everywhere and $f(\partial D) \...
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2answers
58 views

Is there a self-map of the disk with Jacobian everywhere greater than $1$?

It might be silly, but I am not sure how to approach this problem. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Does there exist a smooth map $f:D \to D$ such that $\det df >1$ ...
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59 views

Upper bound on the exact Hausdorff measure

Let the $s$-dimensional Hausdorff measure for a Borel set $F \subset \mathbb{R}^d$ be denoted as follows: $$\mathcal{H}^s(F) := \lim_{\delta \to 0} \mathcal{H}^s_\delta(F) := \lim_{\delta \to 0} \inf ...
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0answers
82 views

Growth rate of the Hausdorff measure

Let the $s$-dimensional Hausdorff measure for a Borel set $F \subset \mathbb{R}^d$ be denoted as follows: $$\mathcal{H}^s(F) := \lim_{\delta \to 0} \mathcal{H}^s_\delta(F) := \lim_{\delta \to 0} \inf ...
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25 views

On the Wulff shape

Let $\phi$ be 1-homogeneous, convex and coercive from $R^n$ to $[0,\infty]$. Define the Wulff Shape associated to $\phi$ as $$ W_{\phi}=\cap_{y\in S^{n-1}}\{ x : x\cdot y<\phi(y) \}. $$ My ...
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32 views

Compatibility of the Hausdorff measure with short exact sequences in normed spaces

Let $(E,\|.\|)$ be a finite dimensional normed space and take $F\subset E$ a subpace, so that we have the canonical short exact sequence $0\rightarrow F\rightarrow^\iota E\rightarrow^\pi E/F\...
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21 views

Integral over the reduced boundary of the measure theoretic outer normal

I am referring to the page 260 of the book "Sets of finite perimeter and geometric measure theory" of F.Maggi: Let $E$ be a set of locally finite perimeter and $H$ an open half space. At some point ...
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23 views

Is the integral of a distribution function on a measure zero set is always zero?

As is well known, the Lebesgue integral of a Lebesgue integrable function on a Lebesgue measure zero set is always zero. Question: Is the integral of a distribution function(distribution means here ...
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32 views

Reference Request: harmonic analysis with Non-Lebesgue reference measure

The Lebesgue measure on $\mathbb{R}^{d}$ admits the following polar decomposition: $$ L(dx) = r^{d-1} dr \lambda(dy), $$ where $\lambda$ is the uniform measure on the Euclidean unit sphere of $\mathbb{...
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2answers
24 views

Hausdorff measure and linear transformation in R^n

Is there a formula of the type $$H^s(TA)=c H^t(A)$$, where $T:R^n\mapsto R^m$ and $s$ is the Hausdorff dimension of $TA=\{Ax\ |\ x\in A\}$ and $t$ is the Hausdorff dimension of $A$?
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219 views

Confusion about the proof of Poincaré lemma for Currents

In Demailly's "Complex Analytic and Differential Geometry" page 20 2.D.4, he proves the Poincaré lemma for Currents. The theorem states as follows: Let $\Omega\subset\mathbb R^m$ be a starshaped open ...
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9 views

Lipschitz domains have finite perimeter?

Is it true that any Lipschitz bounded set in $\mathbb{R}^N$ has a finite perimeter?
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1answer
53 views

A different statement for Fubini's theorem and correcting a solution.

Here is the statement of Fubini's theorem we are using(my professor said that it is from the book of Saks but I searched the book but could not find it, So if anyone knows from which book is this ...
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22 views

Lipschitz domain measurable

Is it true that any Lipschitz domain $\Omega\subset \mathbb{R}^n$ (having the boundary locally the graph of a Lipschitz function) is Lebesgue measurable?
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14 views

Does Lebesgue density theorem holds in Minkowski metric spaces for $0<p<1$?

As pointed out by Heinonen in Lectures on analysis on metric spaces - theorem 1.14 and example 1.15, Besicovitch covering theorem holds in every finite dimensional Banach space and so, in particular, ...
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1answer
42 views

Besicovitch covering theorem in a generic finite dimensional normed vector space

Let $\left(V,\|\cdot\|\right)$ be a finite dimensional normed vector space. If $A \subset V$ denote by $\chi_A$ the indicator function of the set $A$ and if $r>0$ and $v\in V$ denote by $\bar{B}_r(...
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1answer
42 views

Prove that $\mu \times \lambda$ is a measure on $\mathcal{S} \times \mathcal{T}.$

Let $(X, \mathcal{S}, \mu)$ and $(Y, \mathcal{T}, \lambda)$ be $\sigma$-finite measure spaces. If $Q \in \mathcal{S} \times \mathcal{T}$ we define $$\mu \times \lambda(Q) = \int_{X} \lambda(Q_{x})d\...
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1answer
39 views

Does Hausdorff-measurability depend on the choice of Riemannian metric?

Let $M$ be a smooth (second-countable) manifold and let $g, g'$ be smooth Riemannian metrics on $M$ which induce metrics (as in "metric space") $d$ and $d'$ on $M$. Fix $j ≥ 0$ and let $H$ and $H'$ ...
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0answers
41 views

Uniform continuity in Lusin theorem

Let $(X,d)$ be a complete metric space, $\mu$ a finite Borel measure of $(X,d)$ and $f \colon X \to \mathbb{R}$ measurable. Lusin's theorem gives us that for each $\varepsilon>0$ there exists a ...
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169 views

Is it true that Lebesgue's differentiation theorem follows from Lebesgue's density theorem?

Let $(X,d)$ be a separable complete metric space and $\mu$ a probability measure on the Borel subsets of $(X,d)$. Suppose that the Lebesgue's density theorem holds, i.e. that for each Borel set $A$ of ...
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2answers
38 views

One-dimensional equivalent of area in 3 dimensions

We are all familiar with the 3-dimensional notions of volume and area. For example, the volume of a 3-dimensional sphere is given by $V_3 = 4\pi R^3/3$ and its area by $S_3 = 4\pi R^2$, where $R$ is ...
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1answer
76 views

Is a smooth surjective map between disks which is injective a.e. everywhere injective?

Let $0<\lambda$, and let $D_1,D_{\lambda} \subseteq \mathbb{R}^2$ be the closed Euclidean disks of radii $1$ and $\lambda$ respectively. Suppose we have a smooth surjective map $f:D_1 \to D_{\...
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1answer
25 views

Compact set of the plane with boundary of finite 1-Hausdrof meausre as intersection of open set with piewise smooth boundary.

I am considering a compact set $K$ of the plane with $\mathcal H^1(\partial K)=1$. Does there exists open sets with smooth boundary $O_n$, $n\in \mathbb N$ such that $K=\bigcap_n O_n$ and $\mathcal H^...
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0answers
34 views

Application of Area Formula

Let $n,k$ be positive integers satisfying $k\le n$, $\varphi\colon\mathbb{R}^n\to\mathbb{R}$ be a continuous function whose support is compact and $f\colon\mathbb{R} ^k \to \mathbb{R}^n$ be an ...
2
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1answer
199 views

A nearest neighbor conditional probability problem

Let $(\mathcal{X},d)$ a metric space with its Borel $\sigma$-algebra $\mathcal{F}_{\mathcal{X}}$. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $m\in\mathbb{N}$ with $m\...
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0answers
26 views

Why is a regular area-minimizing current necessarily an oriented manifold?

I've been working my way through Leon Simon's geometric measure theory notes and started wondering in how far (or, rather, why) the regular part $\operatorname{reg} T$ of an $n$-current $T \in D_n(\...
4
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1answer
59 views

If $π$ is the projection of a surface $M$ onto the 2-sphere $S^2$, then $σ_{S^2}(B)=\int_{π^{-1}(B)}σ_M(dy)\frac{|⟨ν_M(y),π(y)⟩|}{|y|^2}$

Let $U\subseteq\mathbb R^2$ be open, $\phi:U\to\mathbb R^3$ be an immersion and a topological embedding of $U$ onto $M:=\phi(U)$$^1$, $\nu_M(x)$ denote the unit normal vector of $M$ with $$\det\left({\...

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