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.

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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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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. ...
Daniel Asimov's user avatar
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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? $$|...
Nate River's user avatar
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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 ...
Alex's user avatar
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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-...
dummy's user avatar
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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-...
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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 \...
dlee's user avatar
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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 ...
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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 $\...
Hengchao Chen's user avatar
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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)=\...
Hengchao Chen's user avatar
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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 ...
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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 ...
Nate River's user avatar
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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 ...
No-one's user avatar
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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 ...
Nate River's user avatar
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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 ...
XYC's user avatar
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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 $T(x_0)$ ...
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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} \...
i like math's user avatar
1 vote
1 answer
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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 ...
i like math's user avatar
1 vote
1 answer
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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}{...
i like math's user avatar
3 votes
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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 ...
Leo Moos's user avatar
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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 ...
i like math's user avatar
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. ...
Nate River's user avatar
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1 answer
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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 ...
i like math's user avatar
3 votes
3 answers
266 views

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^...
Nate River's user avatar
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2 answers
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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 \}$ ...
Taraellum's user avatar
3 votes
1 answer
300 views

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....
warzasch's user avatar
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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 ...
Aritro Pathak's user avatar
2 votes
1 answer
264 views

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 ...
No-one's user avatar
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2 answers
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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 ...
No-one's user avatar
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2 votes
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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 $\pi\subset \mathbb{R}^{n+k}$...
No-one's user avatar
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1 vote
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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 ...
Leo Moos's user avatar
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2 votes
1 answer
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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 ...
Leo Moos's user avatar
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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, $\...
Kaira's user avatar
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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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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(...
Sarthak's user avatar
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3 votes
1 answer
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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 $\...
tommy1996q's user avatar
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0 answers
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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 $...
C L 's user avatar
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7 votes
1 answer
366 views

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 ...
Piotr Hajlasz's user avatar
3 votes
0 answers
121 views

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}...
totallyimmersed9's user avatar
3 votes
2 answers
268 views

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 ...
Sarthak's user avatar
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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 ...
Bedovlat's user avatar
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2 votes
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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 ...
Cris.giansu's user avatar
4 votes
0 answers
161 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 ...
Denis Marti's user avatar
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 ...
ru0xffian's user avatar
2 votes
1 answer
117 views

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-...
tommy1996q's user avatar
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}{...
Mikhail Katz's user avatar
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3 votes
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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 ...
hamath's user avatar
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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 ...
Sanae Kochiya's user avatar
2 votes
1 answer
92 views

Mass of the push forward of a k-current with fixed orientation

$\DeclareMathOperator{\Mass}{Mass}$Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smoth map. Given a $2$-vector (in general a $k$-vector but let's stick to $2$) $v_1 \wedge v_2 \in \Lambda_2 (\mathbb{R}^...
tommy1996q's user avatar
3 votes
1 answer
139 views

Do we have uniformization theorems for fractional dimensional spaces?

The Riemann mapping theorem in $\mathbb{R}^2$ is known not to generalize well in higher dimensions and is basically trivial in lower dimensions. I’m interested in how it generalizes for fractional ...
Sidharth Ghoshal's user avatar

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