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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25 views

Convexity of a set of probability densities

Let $Q = $ {$d$-dimensional probability densities with independent marginals}, i.e. $$Q = \left\{ q(x) = \prod_{i=1}^d q_i(x_i)\;\; \big| \;\; x\in \mathbb{R}^d\right\}.$$ I'm wondering if this is ...
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Pointed Measured Gromov-Hausdorff Convergence

It is well-known that one can extend the definition of Gromov-Hausdorff convergence to non-compact metric spaces by instead considering pointed Gromov-Hausdorff convergence. There is also an ...
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A generalization of Green-Gauss divergence theorem to Sobolev functions on sets of finite perimeter

Let $n\ge 2$. Let $u \in W^{1,1}(\mathbb{R}^n)$, i.e. $u \in L^1(\mathbb{R}^n)$ and its distributional gradient is represented by an element of $L^1(\mathbb{R}^n;\mathbb{R}^n)$. By the Sobolev ...
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Measurability of Hausdorff measure under bi-Lipschitz mappings

For $0< s < n$, denote by $\mathscr{H}^s$ the $s$-dimensional Hausdorff measure. Let $E\subseteq \mathbb{R}^n$ be $\mathscr{H}^s$-measurable, and $f:\mathbb{R}^n \to \mathbb{R}^n$ a bi-Lipschitz ...
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Probability measures on the class of nonempty compact subsets

Let $$\mathcal{C}(\Bbb R^n) = \{A \subseteq \Bbb R^n \mid A \neq \emptyset, A \textrm{ compact}\}.$$ I am curious about whether there are some notable probability spaces defined on $\mathcal{C}(\Bbb R^...
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Nested Radon-Nikodym Derivative

Let $\pi$ and $\eta$ be two measures on the same measurable space $(E, \mathcal{E})$ and that $\pi\ll\eta$ so the Radon-Nikodym derivative exists $$ \frac{d \pi}{d \eta} = \rho $$ Assuming all ...
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Smooth function in $C^1$ Sard-type Theorem

In this note, we can use coarea formula to prove the following $C^1$ Sard-type theorem (6.4): Suppose $f:M\to\mathbb{R}^m$, $m<n$, is $C^1$, with $M$ is an $n$-dimensional $C^1$ submanifold of $\...
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Geometric condition number for uniform sampling a real algebraic set

A problem I'm on a hunt to figure out. I'm not sure whether it's better placed here or on Overflow- it has the feeling of being very standard for the right person but I haven't done enough geometric ...
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How to find the diameter of a slice of some annulus in $\mathbb{R}^2$

First excuse my bad drawing. So, Here $\overline{OA} =r,\overline{OB} =2r, \angle{BOE}=\angle{DOE}=\phi, (0<\phi<\pi/2)$. Now I need the 'diameter' of the region shaded in red, where $\...
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25 views

Upper bound of the lower density for a curve-free 1 set

This post is asking for a reference. So in the 'Fractal Geometry' book by K.Falconer, in the chapter 'Local structure of Fractals', there is a statement which says that, If $F$ is a curve free 1-set ...
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Hausdorff dimension of unit closed ball of $ \mathbb{R}^n$ is $n$

Let's admit as definition of Hausdorff dimension the one given by wikipedia https://en.wikipedia.org/wiki/Hausdorff_dimension How to demonstrate dimension of unit closed ball of $\mathbb{R}^n$ is $n$ ?...
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Measure formuation of a conservation law

Let $u$ be a $BV$ entropy-solution of the scalar conservation law $$u_t(x,t) + f(u(x,t))_x = 0$$ It is claimed that then (since the derivatives of a $BV$ function are measures) it holds in the sense ...
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Sobolev spaces with respect to divergence and their properties

Let $n \in \mathbb{N}$, $\Omega$ a non-empty bounded open set of $\mathbb{R}^n$ with Lipschitz boundary and $p \in [1,\infty]$. Define $$V_p:=\bigg\{\overrightarrow{q}\in L^p(\Omega;\mathbb{R}^n) \mid ...
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Why a Lebesgue measure equal to an integral?

For a given face $F$ of $P ∈ \mathcal{K}^n$ be a polytope, the Lebesgue measure of the set $$M_ρ(P, η) ∩ p(P, ·)^{−1}(relint F),$$where $η ∈ \mathcal{B}(Σ)\ (\mathcal{B}(Σ)$ is a $\sigma$-algebra of ...
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Geometrical introduction to Caccioppoli sets

Can anyone suggest me a geometrical introduction to Caccioppoli sets, a.k.a sets of finite perimeter. This Encyclopedia of Mathematics article states a definition of the perimeter of a set in terms of ...
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Hausdorff Dimension of a set in R under the image of Continous map

In Falconer's book there is the following exercize: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function $f(x) = x^2$ and let $F \subset \mathbb{R}$. Show that $\text{dim}f(F) = \text{dim}_H F$....
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Taking Limits in Inequality for Hausdorff Measure

I'm trying to show that when $\mathcal{H}^a(K) > 0$ and $b < a$ then $\mathcal{H}^b(K) = \infty$. I did this by considering a cover $K \subset \{U_i\}$ of radius $\delta$. Then $$\sum|U_i|^a = \...
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Hausdorff Measure of a Compact segment in $\mathbb R$ has infinite hausdorff measure for $s < 1$

This is probably obvious, but I'm having trouble with it. It is an exercise in Falconer: Show that $\mathcal{H}^s([0,1]) = \infty$ if $s \in [0,1)$ I can see that this should loosely be the case: ...
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Show that $\frac1{|\partial B_r(x)|}\int_{\partial B_r(x)}f\xrightarrow{r\to0+}f(x)$

Let $d\in\mathbb N$ and $\sigma_M$ denote the surface measure on $\mathcal B(M)$ for every embedded $C^1$-submanifold $M$ of $\mathbb R^d$. Moreover, let $$\theta_r:\mathbb R^d\to\mathbb R^d\;,\;\;\;x\...
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Definition of Transition Kernel as Operator on Measures

Let $(S, \mathcal{S})$ and $(T, \mathcal{T})$ be two measurable spaces. Let $K: S\times \mathcal{T}\to [0, 1]$ be a Markov Kernel. I read that you can define an operator on measures based on $K$: Let ...
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$\delta$- hausdorff measure of open ball

Let U open ball in $\mathbb{R}^n$, $n \ge 2$, such that diameter $d(U)=\delta$. Let $ 0 \le s \le 1$, we need to prove that $H_{\delta}^s(U)= H_{\delta}^s(\partial U)= H_{\delta}^s(\bar{U}) $. Here $...
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The diameter of Voronoi cells in Euclidean spaces

Let $A \subset \mathbb{R}^d$ and let $(x_n)_{n \in \mathbb{N}} \subset \mathbb{R}^d$ be a sequence dense in $A$. For each $n \in \mathbb{N}$, let $V_{1,n},\dots,V_{n,n}$ the sequence of Voronoi cells ...
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Transform an integral from Hausdorff to Lebesgue

Suppose I have the following integral with respect to the $(N-1)$-dimensional Hausdorff measure $$ \int_{\Gamma} g(x) \,\,d\mathcal{H}^{N-1}(x) $$ where $$ \Gamma = \left\{x\in \mathbb{R}^N\, :\, f(x) ...
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Mixture / Union of Measures defined on disjoint subsets

Suppose I have a partition of $\mathbb{R}^N$ $$ \mathbb{R}^N = \bigcup_{i \in \mathbb{R}} \mathcal{X}_i \qquad \qquad \text{with } \mathcal{X_i} \cap \mathcal{X}_j = \emptyset \text{ if } i \neq j $$ ...
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Scaling Property of Hausdorff Measure

A book I'm reading discusses the proof of the following statement: Let $S$ be a similarity transformation of scale factor $\lambda > 0$. If $F \subset \mathbb{R}^n$, then $$\mathcal{H}^s(S(F)) = \...
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Proof of isoperimetric inequality in $\mathbb R^n$

I am trying to find a proof (using geometric measure theory) of the isoperimetric inequality in $\mathbb R^n,$ see here, but I discover that many proofs online either only tackles the 2-dimensional ...
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Determinantal formula for the volume of a convex hull

Let $d$ be an integer. I will work in $\mathbb{R}^d$. Let $N$ be an integer and $x_0,x_1,\cdots x_N$ be $N+1$ points of $\mathbb{R^d}$. If $N=d$, then I have $d+1$ points and $Vol(Conv(x_0,x_1,\cdots,...
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How to construct injective maps whose differential zig-zags between two matrices?

Let $n,a \in \mathbb{R}^2$ be non-zero vectors. Is there an example for a non-trivial* Lipschitz function $h:\mathbb{R} \to \mathbb{R}$ satisfying $h' \in \{0,1\}$ a.e. such that the map $$u_h(x)=x+h(...
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1answer
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Application of Fubini theorem to a proof of the coarea formula, or why the product of $\mathcal H^{n-m}$ with $\mathcal L^m$ equals $\mathcal L^n$

While reading the proof of the coarea formula in Evans and Gariepy's book, Measure Theory and Fine Properties of Functions, I stumbled upon the following affirmation Let $A\subseteq\mathbb R^n$ be $\...
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Computing $n$-dimensional Lebesgue measure with Euclidean $n$-balls

I am studying the coarea formula proof from Evans and Gariepy's Measure Theory and Fine Properties of Functions. At the start of lemma 3.5, the authors are assuming that we can compute the $n$-...
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Strong convergence vs Weak convergence _ compactness of integral varifolds

I am reading the proof of the compactness of Integral varifolds on L.Simon's book " Lecture on geometric measure theory", there is a part of the proof concerning the conclusion that the ...
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Is the Hausdorff measure absolutely continuous with respect to the lebesgue measure? $H^{d-1} \ll \lambda^d$

I'm sure this is just basic theory but I can't find this ANYWHERE. Is it true that $$ H^{d-1} \ll \lambda^d $$ where $H^{d-1}$ is the $(d-1)$-dimensinal Hausdorff measure and $\lambda^d$ is the $d$-...
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Does this property of Radon-Nykodym derivatives exist

Suppose I have $\pi \ll \lambda$ so that $$ f = \frac{d \pi}{d\lambda} $$ and $\lambda \ll \eta$ so that $$ g = \frac{d \lambda}{d \eta} $$ Can we say that $$ \frac{d \pi}{d\lambda} \frac{d \lambda}{d\...
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Is the line measure operator bounded in $W^{1,p}(\mathbb{R}^2)$

Consider the linear funtional $S_l: W^{1,p}(\mathbb{R}^2) \to \mathbb{R}$ for some segment $l \in \mathbb{R}^2$ such that $S_l (\mu)$ measures the length of $l$ with respect to $\mu dx$. In other ...
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Can I write the integral of a function in terms of its level sets?

I have a function like this $f:\mathbb{R}^2\to\mathbb{R}_+$ $$f(x, y) = e^{-\frac{x^2+y^2}{2}}$$ Its level sets $f(x, y) = c$ are simply circles centered at the origin $$ x^2 + y^2 = \log\left(\frac{1}...
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Inequality for perimeter of Minkowski sum?

This following is a small part of a rather large problem that has been bugging me. Let $\Omega$ be any bounded open set in $\mathbb{R}^{2}$ of finite perimeter. Is it necessarily true that $$\mathcal{...
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Is there a standard terminology for $|f^{-1}(y)|=1$ for almost every $y$?

Let $\Omega \subseteq \mathbb R^n$ be a nice open, connected, bounded subset (say with Lipschitz boundary) and let $f:\Omega \to \mathbb{R}^n$ be a Lipschitz map. Is there a standard terminology for ...
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Relationship between finite Radon measures and bounded variation measures

I cannot figure out which is the relationship between Radon measures and measures that have bounded variation. I think it's almost a matter of definition, but I cannot find a proper source to which ...
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In $\mathbb{R}^{2}$, why does the convex hull have lower perimeter?

One thing I've kind of seen of all over the shop is that in $\mathbb{R}^{2}$ taking the convex hull of a bounded open set whose closure is connected reduces perimeter and increases area. I can see the ...
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Investigating a $1/2$-dimensional sphere

The volume of an Euclidian sphere in $n$-dimensions with radius $r$ equals $$V_n=r^n \frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}.$$ This formula is valid for $n\in\mathbb{N_0}$. Now I am asking myself if ...
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What's the intuition behind the Co-Area formula?

I mainly work in statistics and I know only basic measure theory. I was trying to understand the Co-Area formula by Federer. If $f:\mathbb{R}^M\to \mathbb{R}^N$ is a Lipschitz function with $M \geq N$...
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Not Jordan measurable

How to prove $\forall Q \cap [0,1] \times [0,1]$ is not Jordan measurable? I know as the only rectangle contain by this are points so the inner will be 0, but what about the outter? Is there any way ...
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Density of smooth functions in $L^p$ space on Cantor Set

Let $\mu$ be the Cantor measure on the Cantor set $C$ in $[0,1]$. Is the space of functions obtained by restricting smooth compactly supported functions to $C$ dense in $L^p(\mu)$ for $1 \leq p < \...
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How to show that $D$ is a Borel measurable set and $D_{f}$ is a Borel function.

How to show that $D_{f}$ is a Borel measurable function. Well I have one Lipschitz function $f:\Bbb{R}^{n}\to \Bbb{R}$ and I want to proof that $D_{f}:D\to L(\Bbb{R}^{n},\Bbb{R})$ is Borel function, ...
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Alternative characterization of total variation of $L^1$ functions

This is about exercise 3.3 from Ambrosio, Fusco and Pallara Functions of bounded variation and free discontinuity problems. I struggled with this for some time. Let $\Omega$ be an open set and $u \in ...
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The Existence of the Density of Rectifiable measure

Let $E\subset\mathbb{R}^n$ be a set. Suppose that $0<\mathcal{H}^m(E)<\infty$ and there exist at most countable Lipschits functions $\varphi_i\colon \mathbb{R}^m\to\mathbb{R}^n$ such that $\...
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How complicated can Lipschitz domains be?

A Lipschitz domain $\Omega$ is a domain in $\mathbb R^n$ whose boundary $\partial\Omega$ is locally the graph of a Lipschitz continuous function. For example, any $C^1$ domain is Lipschitz and a cube ...
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Approximation of rectifiable sets

Let us say that I have a $H^d$ measurable, $H^d$ finite set $E \subset \mathbf{R}^n$ such that there exists a sequence of $C^1$ submanifold $(S_i)$ of $\mathbf{R}^n$ such that \begin{equation} H^d(E \...
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1answer
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How many times can $C^{1}$ strictly convex functions intersect on a bounded interval?

The question is related to question here but with a little of modification which makes the task a little difficult Let $f:\mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be two ...
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67 views

Show that an annulus has small Hausdorff measure

Edit: I believe the below (original) problem can be reduced to the following one Let $(X, \rho)$ be a metric space, where $X$ has Hausdorff dimension $d$. Under what conditions on $X$ can we say that ...

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