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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Covering argument for linear elliptic equations

I am looking for some ideas on trying to show if the following sort of result is true or false: Let $u$ be a bounded, local, weak solution of $\text{div} A(x) \nabla u = 0$. Since $u \in W_{loc}^{1,2}...
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Is there an open subset $A$ of $[0,1]^2$ with measure $>\frac{1}{100}$ that satisfies this property?

Can we find for any given $\varepsilon>0$ an open subset $A\subseteq[0,1]^2$ with measure $>\frac{1}{100}$ such that, for any smooth curve $\gamma:[0,1]\to\mathbb{R}^2$ of length $1$, the set $\...
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Riesz's representation of a k-current

Reading "Introduction to GMT" by Simon, at page 136 he says that thanks to Riesz's representation theorem we can view k-currents ar Radon measures, or to be more precise he says that given a ...
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regular analytic 1-dim. foliation of $M=(0,1)^3$

I'm investigating this question: Does there exist a regular analytic $1-$dim. foliation of $M=(0,1)^3$ s.t. all leaves include both $(0,1,1)$ and $(1,0,0)$ or approach these points in the limit? I ...
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What does a hashtag subscript mean?

I've been coming accross notation like this $\gamma = (id,id)_\# \mu$, where the hashtag/pound sign is used in the subscript. From context, it seems to have something to do with marginalizing ...
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Function with harmonic properties

Let $g(z)$ be a continuous function on $\mathbb R^n\setminus \{0\}$. $$ \int_{B_R(0)} |g(z)| dz \leq C_1 $$ for some constant $C_1$, and with $B_R (0)$ being the ball of radius $R$ centered at the ...
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doubling metric measure spaces and Lebesgue differentiation theorem

If $(X,d,\mu)$ is a doubling metric measure space then it known that if $f \in L^1(X)$ is a positive function with $\|f\|_{L^1(X)}=1$ then $$ \lim_{n \to \infty} \frac{1}{\mu(B(x,\frac{1}{n}))}\int_{B(...
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Integration by parts on compact, non-orientable Riemannian manifold with boundary

Let $(M,g)$ be a compact Riemannian manifold, not necessarily orientable or without boundary. Let $\mu$ be a normalized volume measure on $M$ and $u$ be a smooth function on $M$. In some notes that I ...
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Why not relax the definition of perimeter to a.e. $C^1_c$ functions?

We have the following definition of perimeter, freely available from your favourite text on geometric measure theory: $P(E)=\sup\bigg\{\int_E \nabla \cdot \varphi(x)\,dx:\varphi\in C^1_c(\mathbb{R}^n;\...
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Relationship between TOWER property and Disintegration of a measure.

What is the relationship between the Law of Total Expectation and the Disintegration Theorem? I am interested both in the general case $\mathbb{E}[\mathbb{E}[X \mid \mathcal{G}] \mid ]\mathcal{H}] = \...
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How to write conditional expectation as integral with respect to regular conditional distribution

Can I write a conditional expectation as an integral with respect to a regular conditional distribution? Set Up $(\Omega, \mathcal{F}, \mathbb{P})$ probability space $(\mathsf{E}, \mathcal{E})$ ...
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Regular Conditional Probability vs Regular Conditional Distribution

Wikipedia (and different books too) seem to give two different definitions of what a regular conditional probability is. What is the correct definition and how do they relate? It seems to me that the ...
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Existence of non-volume preserving automorphisms of normal distribution

Let $X=\mathbb{R}^d$, let $p$ the standard normal distribution on $X$, with zero mean and identity covariance, and let $f:X \to X$ be a diffeomorphism that preserves this normal distribution. For ...
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Geometry of Voronoi cells of $n+1$ equidistant points in $\mathbb{R}^n$

These questions come from the reading of the following article: Milman, E., & Neeman, J. (2022). The Gaussian Double-Bubble and Multi-Bubble Conjectures. Annals of Mathematics, 195(1), 89-206. (...
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Outward pointing vector on Lipschitz boundary

I have some questions on Lipschitz domains and their unit outward pointing vectors. My questions are listed below, I would appreciate direct answers and/or references on the subject. What is the good ...
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Why we need a Hausdorff measure here?

If $\varphi\in \mathcal{C}^1_c(A)$ and $G\in \mathcal{C}^1(A,\mathbb{R}^n)$, then $\int_A \text{div} (\varphi G(x))\ dx=0$. ($K$ is compact support) Proof: $I=\int_B \text{div} (\varphi \ G(x))\ dx=\...
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Radon-Nikodym derivative of pushforward of Lebesgue measure by differentiable function with respect to Lebesgue measure

Here is the set up: $f:\mathbb{R}^n\to\mathbb{R}^m$ is a measurable function $\lambda^n$ is the $n$-dimensional Lebesgue measure $f_*\lambda^n$ is the pushforward of $\lambda^n$ by $f$ $f_*\lambda^n \...
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Relationship between Disintegration Theorem and Co-Area formula

I have a feeling the Disintegration Theorem and the Co-Area formula for Lipschitz functions are actually very much related but I cannot seem to formalize it. DISINTEGRATION THEOREM: Let $(\mathsf{X}, ...
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Is every radon-nikodym derivative a random variable?

Let $(\mathsf{X}, \mathcal{X}, \mu)$ be a measure space with $\mu$ begin $\sigma$-finite. Definition of Random Variable: Let $(\mathsf{Y}, \mathcal{Y})$ be a measurable space. A function $\xi:\mathsf{...
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How is a restricted measure defined over sets not in sub-sigma algebra?

According to ProofWiki given a measure space $(\mathsf{X}, \mathcal{X}, \mu)$ and a subsigma algebra $\mathcal{Y}\subseteq\mathcal{X}$, the restriction of $\mu$ to $\mathcal{Y}$ is the measure $\mu\...
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Are these definitions of dimension for a real algebraic variety all equivalent?

I would like to believe that the following notions of dimension are all equivalent for an a real algebraic variety in $\mathbb{R}^n$. Lower Minkowski dimension Upper Minkowski dimension Hausdorff ...
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Radon-Nikodym derivative of Lebesgue measure with respect to Hausdorff measure is Jacobian term?

Consider the measurable space $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ with Lebesgue measure $\lambda$. I have a submanifold of $\mathbb{R}^n$ defined as the level set of a smooth function $f:\...
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Property holding almost everywhere implies existence of dense subset?

I'm working through some literature from Geometric Measure Theory for an assignment paper and have got stuck at a step in the proof of the Federer-Volpert theorem. The reasoning is as follows: ...
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Different definitions of Hausdorff measure (balls vs cubes vs arbitrary open sets)

The $d$-dimensional Hausdorff (outer) measure of a set $A ⊆ \mathbb{R}^n$ is usually defined to be $$ \mathcal{H}^d(A) = \lim_{δ \to 0} \inf \left\{ \sum_i \operatorname{diam}(U_i)^d ~\middle|~ A ⊆ \...
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Prove that if $\mathcal H^1(F) = 0$ for $F\subset \Bbb R$, then $\Bbb R\setminus F$ is dense

Prove that if $\mathcal H^1(F) = 0$ for $F\subset \Bbb R$, then $\Bbb R\setminus F$ is dense. I need to prove the above result to understand Proposition $3.5.$ in Falconer's Fractal Geometry. $\...
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Cantor-like functions for $\xi\neq \frac{1}{3}$

Let $\xi \in (0,\frac{1}{2})$. Let $C_\xi$ be the perfect symmetric set built by iterating the transformation $$[0,1] \to [0,\xi]\cup [1-\xi, 1].$$ The set $C_\frac{1}{3}$ would then correspond to the ...
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Weak limit of a certain measure sequence is stationary

I am not very conversant about this topic and was given this question as an assignment. This is my first post so I do apologize if the question is very silly. Let $\mu$ be any probability measure on a ...
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A lattice acts in a locally compact group

I'm taking a topics course and trying to fill in details from a lecture. I'm hoping someone can help fill things in or point to a resource where these basics are covered. We have a lattice $\Lambda &...
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Almgren and Taylor's movie about soap bubbles

As I know, in 1970's two well-known geometric analysts, Fred Almgren and Jean Taylor, along with mathematician Michele Emmer, produced a film about minimal surfaces entitled "Soap Bubbles". ...
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Understanding the notion of approximate tangent space, with examples

I am studying geometric measure theory, and I am having some trouble understanding how to deal with approximate tangent spaces. I would like an example/exhibition of an approximate tangent space in a ...
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A function of bounded variation in a regular set has bounded variation in $\mathbb R^N$ and a formula for its variation

Let $\Omega \subset \mathbb R^N$ be an open set. The total variation of a function $u \in L^1(\Omega)$ in $\Omega$ is given by $$ |Du|(\Omega) := \sup\left\{\int_\Omega u \ \text{div}\varphi \ dx \ : \...
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Reduction to parallelepipeds with dyadic projections

I'm working with a covering lemma by Córdoba that states: "Given a parallelepiped $R \subset \mathbb{R}^3$ with sides parallel to the coordinate axes, denote its side lenghts by $x_R, y_R$ and $...
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2 votes
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An elementary proof that finite perimeter implies finite volume (up to complementation)

While I was studying some geometric measure theory I came across the following fact: Let $E \subset \mathbb{R}^n $ be a set of finite perimeter $P(E, \mathbb{R}^n)<+\infty$, prove that $E$ or $E^c$ ...
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Decomposing measures as products Fubini style

Let $B \subset \mathbb R^n$ be the unit ball centered at the origin, $I = [0,1] \subset \mathbb R$, and $C=B\times I\subset \mathbb R^{n+1}$ be a cylinder. Let $\chi_C$ be a the indicator function of ...
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How to understand the coarea formula for homotopies

I am trying to understand a claim in a paper https://arxiv.org/pdf/1303.7427.pdf (Observation 4.2) that the integral over the Jacobi-determinant of a regular homotopy $H:[0,1]\times [0,1]\rightarrow \...
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3 votes
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A detail in a proof of the Steiner's inequality about sets of finite perimeter

Given an open limited set $E$ of $\mathbb{R}^n$ with smooth boundary, how could I subdivide this set $E$ in a finite number of normal sets? I remind you that the normal set is a set of this form, ...
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3 answers
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Is it possible to calculate sphere surface area with circles?

Imagine cutting a sphere into circles(the distance between the two circles is almost zero). Then is it correct to say that the sum of the circumference of all the circles is the surface area of the ...
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Area of a spherical cap for isotropic measures

A measure $\mu$ on $S^{n-1}$ is called isotropic if $$\int_{S^{n-1}} \langle \theta, x \rangle^2 d\mu(x)=\frac{\mu\left({S^{n-1}}\right)}{n}$$ for all $\theta\in S^{n-1}$. I want to know if there are ...
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1 vote
1 answer
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Points of density

Let $x \in \mathbb{R}^n$, $r_0 > 0$ and $\lbrace C_r \rbrace_{0 < r < r_0}$ be a family of Borel sets such that, for some $\beta > \alpha > 0$, $B(x,\alpha r) \subset C_r \subset B(x, \...
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measure to establish closeness of step functions to some curve

I wonder, if some mathematician could help me please? I have this scenario (please ignore black lines): I would like to have a metric between +1 and -1 indicating how far the curves (step functions) ...
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If a distribution and its gradient have the same finite order then the order must be zero

As the title suggests, I would like to know whether it is possibile for a distribution and its (distributional) gradient to have the same finite and greater or equal than one order.
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Are the Besicovitch Covering Constant known exactly?

My question above summarizes my question. I have been studying geometric measure theory and encountered the Besicovitch covering lemma. Also an auxiliary result "Fix an arbitrary set $A \subset X$...
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Does the following operation preserve Lipschitz continuity?

For fixed $r>0$, consider $\mathcal{L}: L^\infty(0,1) \rightarrow L^\infty(0,1)$, with $(\mathcal{L}f)(x) = \|f\|_{L^\infty(B_r(x))}$. If $f$ is Lipschitz continuous, is $\mathcal{L}f$ Lipschitz ...
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Change of variable formula for Haar measure on product of Lie Groups

First let me recall the usual change of variable formula: Let $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ be a bijection which is Frechet differentiable, $U\in\mathrm{Open}(\mathbb{R}^n)$, and $f:\mathbb{R}^...
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Does non-zero and finite Hausdorff dimension imply non-zero and finite Hausdorff measure?

Is it possible that a set $A$ has Hausdorff dimension $d=\mathrm{dim}_H (A)\in (0, \infty) $ but $ H^d (A)\notin (0, \infty) $? In other words, positive and finite Hausdorff dimension but its ...
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1 vote
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Tangent measure to $\delta_y$.

I don't know anything about geometric measure theory but I was watching this video to try to understand the concept of tangent measures. I read that if $\mu=\sum c_i\delta_{y_i}$, then $Tan(\mu,a)=\{c\...
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3 votes
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Covering with sets of negligible boundary

I am studying causality theory in Lorentzian length spaces, and I have a question about geometric measure theory in general (it will help me with a proof I am trying to finish): Suppose we have a ...
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9 votes
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Can I split $E$ in equal volume parts?

Problem: Let $E \subset \mathbb{R}^N$ be a connected, bounded, open and smooth (or just $N$-measurable) set and denote with $\mathcal{L}^N$ the Lebesgue measure. Define $\Omega_i = \{ x \in \mathbb{R}^...
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1 answer
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De Giorgi's structure theorem for reduced boundary - why compact submanifolds?

De Giorgi's structure theorem states the following (Theorem 15.9 from Maggi's Sets of Finite Perimeter and Geometric Variational Problems, or e.g. Theorem 4.3 of Giusti's Minimal Surfaces and ...
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Exact, Closed and Harmonic Currents on Compact Complex Manifolds and their Quotient Spaces

Let $M$ be a compact complex manifold of complex dimension $m$. It is well-known that, since every exact current is closed, we can define $$D_{current}^p(M)=\frac{\{Closed p-currents\}}{\{Exact p-...
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