Questions tagged [geometric-measure-theory]
Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.
715
questions
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Hahn-Mazurkiewicz with finite one-dimensional Hausdorff measure
Suppose that there is a continuous surjection from $[0,1]$ to a metric space $(X,d)$. If $(X,d)$ has finite one-dimensional Hausdorff measure, must there exist a Lipschitz surjection from $[0,1]$ to $(...
6
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1
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198
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Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.
Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
2
votes
1
answer
101
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On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
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0
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91
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Are these two spaces of functions identical?
Let the function space $A$ denote all functions $f : [0, 1) \to [0, 1)$ such that, for some set $Z$ of Lebesgue measure zero, the derivative $f'$ exists on $[0, 1) \setminus Z$ and $|f'| = 1$ there.
...
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68
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Solution to the Eikonal equation with almost everywhere continuous derivative
Let $\Omega$ be an open, bounded, connected subset of $\mathbb R^n$ with smooth boundary. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following system of PDE?
$$|...
2
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0
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177
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Statistical invariants of Riemannian manifolds
$\DeclareMathOperator\diam{diam}\DeclareMathOperator\rad{rad}\DeclareMathOperator\iso{iso}\DeclareMathOperator\com{com}\DeclareMathOperator\con{con}$A cheap way of defining invariants of Riemannian ...
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29
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On the Lipschitz parametrizability of polynomials of fixed Mahler measure
Background
For a polynomial $f(x) = a(x-\alpha_1) \cdots (x - \alpha_n) \in \mathbb{C}[x]$, its Mahler measure is defined to be
$$M(f) = |a| \prod_{i=1}^n \max\{1, |\alpha_i|\}$$
In Lemma 1, Masser-...
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84
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Why does $\omega$ belong to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-2 + n\varepsilon^3}$ different $\theta$?
In this paper, there is the following claim (Pg. 1850):
If $1 - \eta(w) \ne 0$, then $|\omega| \ge \rho$. In that case, $\omega$ belongs to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-...
2
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80
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Equivalence class of parametrized surfaces which induce the same current
Suppose $M$ is a smooth manifold of dimension $n \geq 2$. A $k$-current is a linear functional on compactly supported smooth forms on $M$, denoted $T: \Omega^k_c(M) \to \mathbb{R}$.
Let $X: [0,1]^2 \...
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Proof that, for $u \in H^1$, $\{ u > \alpha \}$ is a quasi open set
I am reading the monograph by A. Henrot, Extremum problems for eigenvalues of elliptic operators. In chapter 2, the notion of a quasi-open set is defined (see the relevant definitions at the end of ...
1
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0
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79
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Measurability of the union of cut loci along a curve
Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
1
vote
1
answer
148
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Differentiability of an integral of geodesic distance
Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$.
Q1: Define
$$
g(t)=\...
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0
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89
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Bounding the area of the image of a set by product of maximum of lengths
Let $F:[0,1]\times[0,1]\to \mathbb {R}^2$ be a smooth function. Given $x\in [0,1]$, let $\ell_x:=\{x\}\times [0,1]$, and given $y\in [0,1]$, let $\ell_y:=[0,1]\times \{y\}$.
My question feels ...
4
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1
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171
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What are the possible blow up limits of an $L^1$ function?
Let $f: [0, 1] \to \mathbb R$ be an $L^1$ function. Define for each $r > 0$, the blow up $f_r:[0, 1] \to \mathbb R$ by
$$f_r (x) := \frac{f(rx)}{r}.$$
Suppose $f_r$ converges in $L^1$ to some ...
2
votes
0
answers
137
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$\mathscr{H}^{n-2}(\Sigma)< \infty$ implies $\mathscr{H}^{n-1}(\pi(\Sigma))=0$
Let $\Sigma\subset \mathbb{R}^{n+1}$ be a set with $(n-2)$-dimensional Hausdorff measure finite, i.e. $\mathscr{H}^{n-2}(\Sigma)<\infty$. Let $\pi:\mathbb{R}^{n+1}\to \mathbb{R}^n$ be the ...
2
votes
1
answer
179
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Macroscopic sets - a notion of largeness for Lebesgue null sets
Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
1
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81
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Periodic orbits in planar smooth billiard table with large periods
Given a plane billiard table with a smooth boundary which is a Jordan curve, I wonder if there is always a periodic orbit with sufficiently large period.
Formulation of my question: We are considering ...
0
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85
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When can an affine functional on the dual be represented as an element of a Banach space?
In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
1
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2
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111
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Computation of tangent functional
In Measures Which Agree on Balls by Hoffmann-Jørgenson, the tangent functional is defined as follows.
If $x \in S$, we define the tangent functional $\tau(x,\cdot)$ at $x$ as
\begin{equation}
\...
1
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1
answer
182
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Gateaux differentiability of the norm in Banach spaces
I'm struggling to understand a particular implication in the proof of Corollary 5 of this paper involving Gateaux differentiability of the norm. The claim is that Gateaux differentiability of the norm ...
1
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1
answer
82
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Potentially elementary question on affine functions on Banach spaces
In Measures Which Agree On Balls by Hoffmann-Jørgensen, it is claimed that the function defined on $T(x)$, the set of normals to the unit sphere at $x$, given by
$ \varphi(x^*) = \left\{
\begin{array}{...
3
votes
1
answer
128
views
Is every area-minimizing cone a level set of a least-gradient function?
Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be a minimizing cone with an isolated singularity. One example, in a space of even dimension, i.e. if $\mathbf{R}^{n+1} = \mathbf{R}^{2m}$, is the Simons ...
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votes
1
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122
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What is an "open Baire set"?
In Measures Which Agree on Balls by Hoffmann-Jørgensen, it is stated that if $\varphi$ is a Baire function (which I presume means a pointwise limit of continuous functions), then $\{a<\varphi\}$ is ...
22
votes
1
answer
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A gerrymandering problem - can you always turn a tie into a landslide victory?
Note: Here we use $|A|$ to denote the Lebesgue measure of a measurable subset $A$ of $\mathbb R^2$.
Your party is running for election! In your country, voters are approximately uniformly distributed. ...
1
vote
1
answer
158
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Definition and properties of tangent functional
I am reading Measures Which Agree on Balls by Hoffmann-Jørgensen and I am somewhat confused. Here, $E$ is a Banach space, $S$ is the unit sphere, and $x \in S$.
We let $\tau(x, \cdot)$ denote the ...
3
votes
3
answers
267
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Extending a $C^1$ function on $\mathbb R^n$ to a set of finite $\mathcal H^{n-2}$ measure
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure.
Let $n \geq 2$ be an integer, and $E \subset \mathbb R^n$ be a set of finite $\mathcal H^{n-2}$ measure.
Suppose $f: \mathbb R^...
3
votes
2
answers
170
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Flat norm of currents and minimal surfaces
Let $A$ be a $k \leq n$ integral current with compact support over $\mathbb{R}^n$ (for conciseness).
Its flat norm $F(A)$ can be defined via
$ F(A) = \inf \{ M(T) + M(S) \, | A = T + \partial S \}$
...
3
votes
1
answer
301
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In which ways did geometric flows and variational methods from Riemannian geometry enter the symplectic world?
I am interested to learn about the role of geometric analytic methods for solving problems in symplectic geometry, In particular, I would like to know what results heavily rely on this machinery (incl....
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0
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80
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Weak geometric lemma and Ahlfors-David regular boundary of a domain
It has been conjectured by David and Mayboroda (in the paper 'Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions', while proving theorem 6.1 in the ...
2
votes
1
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266
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If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected
Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
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2
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If $\mathcal{H}^{n-1}(K)=0$ then $\mathcal{H}^n(K\times \mathbb{R})=0$
I am reading a paper Simon and Wickramasekera - A Frequency Function and Singular Set Bounds for Branched Minimal Immersions where the authors seem to claim that if $K\subset\mathbb{R}^n$ is a compact ...
2
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0
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162
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A question from Leon Simon's "Lectures on Geometric Measure Theory"
In a book I am reading (Leon Simon, Lectures on Geometric Measure Theory) at some point the author claims that if a certain property $(P)$ holds for almost every $n$-plane $\pi\subset \mathbb{R}^{n+k}$...
1
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0
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53
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Limits of branched minimal immersions into the sphere
Can a sequence of branched minimal immersions $M_j^n$ in the round sphere $S^{n+1}$ converge to a smoothly embedded $\Sigma$, in the sense that $ M_j \to 2 \Sigma$ as currents or varifolds?
The case ...
2
votes
1
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132
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Minimal graph with confusing (?) property
Let $n \geq 2$ and $C = \{ (x,y) \in \mathbf{R}^{2n} \mid \lvert x \rvert = \lvert y \rvert \} \subset \mathbf{R}^{2n}$ be the Simons cone. (Whether this is area-minimizing or not does not seem to ...
0
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0
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123
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What does $g_x^{-1}$ mean where $g$ is a Riemannian metric?
I am reading this paper about the Wasserstein-Fisher-Rao distance and they define the Wasserstein-Fisher-Rao distance on a manifold as follows (page 9):
Here, $M$ is a compact Riemannian manifold, $\...
1
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0
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81
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Symmetry of the isoperimetric profile
Given a probability measure $\mu$ on a metric space $(X, \mathsf{d})$, the $(\mu-)$Minkowski content of a set $A$ is defined as
$$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
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1
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Riesz energy for open sets in dimension $1$
This is a continuation of the question Calculation of Riesz energy for balls . As there are three questions,;I am posting a new question here. Riesz energy for a ball $B(x_0,r)$ is given by
$$I_s(B(...
3
votes
1
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117
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Given a set of finite perimeter $\Omega$ s.t. $\partial ^* \Omega =\partial \Omega$, it's not true that $P(\Omega)= \mathcal{H}^{n-1} (\Omega)$
In the article "Funzioni BV e tracce" by Anzellotti and Giaquinta, at page 6 you can read (assume $\Omega \subset \mathbb{R}^n$ open): "The following example shows that the hypothesis $\...
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0
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133
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Properties of doubling metric spaces
At present I work with tools that involves doubling metric space, my definition of DME is:
A metric space $X$ is called doubling with constant $N$, where $N \geq 1$ is an integer, if, for each ball $...
7
votes
1
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366
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Exotic homeomorphisms of a cube
If $\varphi:\mathbb{R}\to\mathbb{R}$ is continuous, non-constant, non-decreasing, and differentiable a.e. with $\varphi'=0$ a.e., then the mapping
$$
\Phi(x,y)=(x+\varphi(x),y+\varphi(y))
$$
is a ...
3
votes
0
answers
122
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Convergence of the perimeter of level sets
I have already posted this question on Math StackExchange. Suppose you have a sequence of $C^1$ functions $\{\phi_n\}_{n\in \mathbb{N}}$ defined on $\mathbb{R}^n$ that converges in $C^{1}_{\mathrm{loc}...
3
votes
2
answers
271
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Calculation of Riesz energy for balls
I was reading stuffs about Riesz energy which is defined for an open subset $U\in\mathbb{R}^d$ by $I_s(U)=\int_U\int_U|x-y|^{-s}\ dx\ dy$ where $dx$ and $dy$ are Lebesgue measure in $U$. Now if I take ...
1
vote
0
answers
118
views
Absolute continuity of the volume growth in a metric space
Let $(M,d)$ be a metric space (separable, complete, better?) and let $\mu$ be a ($\sigma$-additive, positive, locally finite, regular?) Borel measure on $M$. For $x\in M$ consider the volume growth ...
2
votes
0
answers
86
views
Tangent cones at infinity and the regularity of minimal submanifolds
In the famous paper by D. Fischer-Colbrie "Some rigidity theorems for minimal submanifolds of the sphere", the very first sentence reads: It is well known that the regularity of minimal ...
4
votes
0
answers
162
views
Finding balls with big measure
Let $(X,d)$ be a compact metric space $n \in \mathbb{N}$ and $\mu$ a finite Borel measure. Suppose there exists $\delta, R>0$ such that for all $0<r<R$.
$$\mu(B(x,r)) < \delta r^n.$$
Under ...
1
vote
0
answers
56
views
Cardinality of intersections of lines with irregular 1-sets in the plane
From Falconer's book (The geometry of fractal sets), Lemma 3.2 says that the intersection of irregular 1-sets with straight lines is of zero $H^1$ measure. What do we know about the cardinality of ...
2
votes
1
answer
119
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Definition of integral over level sets in coarea formula
This is probably a simple question, maybe more suited for MSE. In the coarea formula, you have
$$\int_{{\mathbb{R}}^n} g (x) |\nabla f(x)|\, dx= \int_\mathbb{R} \left(\int_{\{f=t\}} g d \mathcal{H}^{n-...
5
votes
0
answers
340
views
Extending Gromov's inequality
In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound
$$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol}
\frac{\stsys_2^n}{...
3
votes
0
answers
78
views
Excess function and mean curvature
I'm reading Savin's lecture notes on nonlocal minimal surfaces (available here) and he defines what he calls the excess function of a smooth set $E$ with $0\in\partial E$ by$$e(r)=\frac{\int_{\partial ...
0
votes
0
answers
78
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Purely non-atomic measure on the Gromov boundary of a finitely generated free group
In the set-up of my previous post, let $\theta$ be a purely non-atomic probability regular measure defined on the Borel $\sigma$-algebra of the metric space $(\partial F, d)$. We say $\theta$ admits a ...