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.
685
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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
$$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
- 634
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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
$$I_s(B(...
- 31
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1
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Given a set of finite perimeter $\Omega$ s.t. $\partial ^* \Omega =\partial \Omega$, it's not true that $P(\Omega)= \mathcal{H}^{n-1} (\Omega)$
In the article "Funzioni BV e tracce" by Anzellotti and Giaquinta, at page 6 you can read (assume $\Omega \subset \mathbb{R}^n$ open): "The following example shows that the hypothesis $\...
- 383
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Properties of doubling metric spaces
At present I work with tools that involves doubling metric space, my definition of DME is:
A metric space $X$ is called doubling with constant $N$, where $N \geq 1$ is an integer, if, for each ball $...
- 1
7
votes
1
answer
356
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Exotic homeomorphisms of a cube
If $\varphi:\mathbb{R}\to\mathbb{R}$ is continuous, non-constant, non-decreasing, and differentiable a.e. with $\varphi'=0$ a.e., then the mapping
$$
\Phi(x,y)=(x+\varphi(x),y+\varphi(y))
$$
is a ...
- 25.7k
3
votes
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106
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Convergence of the perimeter of level sets
I have already posted this question on Math StackExchange. Suppose you have a sequence of $C^1$ functions $\{\phi_n\}_{n\in \mathbb{N}}$ defined on $\mathbb{R}^n$ that converges in $C^{1}_{\mathrm{loc}...
3
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2
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Calculation of Riesz energy for balls
I was reading stuffs about Riesz energy which is defined for an open subset $U\in\mathbb{R}^d$ by $I_s(U)=\int_U\int_U|x-y|^{-s}\ dx\ dy$ where $dx$ and $dy$ are Lebesgue measure in $U$. Now if I take ...
- 31
1
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105
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Absolute continuity of the volume growth in a metric space
Let $(M,d)$ be a metric space (separable, complete, better?) and let $\mu$ be a ($\sigma$-additive, positive, locally finite, regular?) Borel measure on $M$. For $x\in M$ consider the volume growth ...
- 1,797
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51
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Tangent cones at infinity and the regularity of minimal submanifolds
In the famous paper by D. Fischer-Colbrie "Some rigidity theorems for minimal submanifolds of the sphere", the very first sentence reads: It is well known that the regularity of minimal ...
- 21
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150
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Finding balls with big measure
Let $(X,d)$ be a compact metric space $n \in \mathbb{N}$ and $\mu$ a finite Borel measure. Suppose there exists $\delta, R>0$ such that for all $0<r<R$.
$$\mu(B(x,r)) < \delta r^n.$$
Under ...
- 41
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Cardinality of intersections of lines with irregular 1-sets in the plane
From Falconer's book (The geometry of fractal sets), Lemma 3.2 says that the intersection of irregular 1-sets with straight lines is of zero $H^1$ measure. What do we know about the cardinality of ...
- 11
2
votes
1
answer
96
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Definition of integral over level sets in coarea formula
This is probably a simple question, maybe more suited for MSE. In the coarea formula, you have
$$\int_{{\mathbb{R}}^n} g (x) |\nabla f(x)|\, dx= \int_\mathbb{R} \left(\int_{\{f=t\}} g d \mathcal{H}^{n-...
- 383
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Extending Gromov's inequality
In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound
$$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol}
\frac{\stsys_2^n}{...
- 14.1k
3
votes
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72
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Excess function and mean curvature
I'm reading Savin's lecture notes on nonlocal minimal surfaces (available here) and he defines what he calls the excess function of a smooth set $E$ with $0\in\partial E$ by$$e(r)=\frac{\int_{\partial ...
- 121
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76
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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 ...
- 167
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votes
1
answer
75
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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}^...
- 383
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votes
1
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131
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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 ...
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119
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What's the derivative of the characteristic function of the intersection of two Caccioppoli sets?
Suppose that $\Omega \subset \mathbb{R}^n$ is a bounded domain. A set $E\subset \mathbb{R}^n$ is called a Caccioppoli set if $\sup\left\{ \int_{\Omega'} \chi_{E} \operatorname{div}(X) \, dx\right\}&...
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Plateau problem for fluxes of curves
Consider $\mathbb{R}_t \times \mathbb{R}_x ^n$ , let $b_1(t,x)$ and $b_2 (t,x)$ be two velocity fields with all the regularity you want and consider the flow of the point $(0,x_0)$ for a time $T$. ...
- 383
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602
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Use of Stein's maximal principle in Bourgain's paper on Besicovitch sets
I'm trying to understand Bourgain's paper "Besicovitch type maximal operators and applications to Fourier analysis". Let $\xi\in S^2\subset\mathbb{R}^3$ be a unit vector and $\delta>0$, ...
- 255
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1
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95
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Singularities of mean-convex MCF in the sphere?
Let $\Sigma^n \subset S^{n+1}$ be a codimension one, embedded minimal hypersurface in the sphere. As the sphere has positive Ricci curvature, this must be unstable. In particular, perturbing $\Sigma$ ...
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votes
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If a function $f$ is $\varepsilon$-times Lebesgue differentiable, is $f$ continuous?
Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. Given an $\varepsilon > 0$, we say $f$ is $\varepsilon$-times Lebesgue differentiable if
$$\lim_{r \to 0} \frac{\int_{B_r (x)} |...
- 3,414
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42
views
Intersection of $n$-dimensional minimal surfaces with two-dimensional planes
Let $M^n \subset \mathbf{R}^{n+k}$ be a smoothly embedded minimal surface. When the dimension is $n = 2$ and the codimension is $k = 1$ the intersection of $M$ with planes is well understood. If $M$ ...
- 4,626
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78
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How to find a smallest parallelepiped that bounds the unit ball in a normed space
Consider a finite dimensional normed space $(V,\Vert \cdot \Vert)$. How to find a basis $(e_i)$ of $V$ such that the unit closed ball $\overline B_1$ centered at $0$ is contained in $ P:= \{ x \in V : ...
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114
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Crofton formula: expected intersections is to length as variance is to what?
There is this beautiful Crofton formula for the length $L(C)$ of
a curve $C$ on the round unit 2-sphere: you take the expected number
of intersections of $C$ with a random great circle and multiply
by ...
- 6,885
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votes
1
answer
117
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Why is there a $\mathcal{H}^d$-null set in the definition of d-rectifiable set?
Given a set $A \subset \mathbb{R}^n$, this is called d-rectifiable if it can be covered by a countable union of images of lipshitz functions from $\mathbb{R}^d $ to $ \mathbb{R}^n $ and a $\mathcal{H}^...
- 383
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Are the isoparametric cones the only known examples of minimizing hypercones?
After searching the literature for a long time, it seems to me that the only known examples of area minimizing hypercones are isoparametric cones and their products with $\Bbb R^k$. By isoparametric ...
- 131
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93
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Varifold convergence of images of Sobolev maps
Suppose I have a sequence of maps $\{f_k:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^{n+1}\}$ such that:
$f_k\rightharpoonup f_*$ weakly in $W^{1,p}(\Omega,\mathbb{R}^{n+1})$,
The images $\Sigma_k:...
- 151
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157
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Defining area / n-volume of a finite metric space
Let $(X, d)$ be a finite metric space. I've seen several answers to the question when can $X$ be isometrically embedded into Euclidean space (or, more generally, Riemannian manifold). I'm interested ...
- 748
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votes
1
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Generalization of a bounded variation
Let $(X, d)$ be a metric space. We will say that $\gamma \colon [a,b] \to X$ is of bounded variation, if
\begin{equation}
V(\gamma) = \sup_{a=t_0 < \cdots < t_n < b} \sum_{i=1}^n d( \gamma(...
- 748
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votes
1
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51
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If $ \mathcal{H}^k(B_1(0)\cap S)\leq A\omega_k $ when $ \mathcal{H}^k(B_r(x)\cap S)\leq A\omega_kr^k $ for all $ 0<r<\delta $, $ x\in\mathbb{R}^n $?
Let $ S\subset\mathbb{R}^n $ is of finite $ k $-dimensional Hausdorff and $ 0<\delta<1 $ is a constant. If for any $ x\in\mathbb{R}^n $ and $ r>0 $, we hae
$$
\mathcal{H}^k(S\cap B_r(x))\leq ...
- 1,049
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votes
0
answers
183
views
Sweeping out the disk: what comes out?
In 2008, Larry Guth gave a new proof of a theorem of Gromov about the min-max widths of the unit $n$-ball. This states that the $p$-parameter width $\omega_p(k,n)$ (of sweepouts with $k$-dimensional ...
- 4,626
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98
views
Plateau problem in the disk: a question about geodesic nets
Consider given a finite collection of points along the boundary of the unit disk $D \subset \mathbf{R}^2$:
\begin{equation}
p_1,\dots,p_{2n} \in \partial D.
\end{equation}
We assume that these are all ...
- 4,626
7
votes
2
answers
386
views
Vector measures as metric currents
Currents in metric spaces were introduced by Ambrosio and Kirchheim in 2000 as a generalization of currents in euclidean spaces. Very roughly, a principle idea is to replace smooth test functions (and ...
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answers
121
views
What prevents spontaneous oscillations in minimal surfaces?
Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be an unstable minimal cone with an isolated singularity at the origin. Let $\Sigma \subset \partial B$ be its link, and $(\varphi_i)$ be the eigenfunctions ...
- 4,626
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votes
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answer
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Exceptional set for Marstrand's projection theorem
If $A\subset\mathbb{R}^2$ is a Borel measurable set and $p_\theta$ is projection onto the line spanned by $(\cos\theta,\sin\theta)$, then it is well known that for almost every $\theta\in[0,2\pi]$, $...
- 255
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2
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Calculating the Fourier dimension of a real interval $\left[a, b\right]$
(Preliminaries:) 1.) Let $S\subset\mathbb{R}^n$ and define $\mathcal{M}(S) = \{\text{$\mu$ a Borel measure}: \text{$0 < \mu(S) < \infty$ and $\mathrm{support}(\mu)\subset S$}\}$.
2.) Define the ...
- 121
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0
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70
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Reference request: theory for local minimizers in the calculus of variations
Let $F: \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be the Lagrangian. We say that $f \in X$ is a local minimizer of the variational integral if for all compact sets $C \subset \...
- 111
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0
answers
54
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Defining minimality 'through deformations'
Let $U \subset \mathbf{R}^{n+k}$ be a bounded open set, and $T \in \mathbf{I}_n(U)$ be an $n$-dimensional integral rectifiable current. Say that $T$ is stationary through (homological) deformations if ...
- 4,626
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Singular asymptotic limits of mean-convex MCF
Let $(M_t \mid t \geq 0)$ be a mean-convex mean curvature flow of hypersurfaces in ambient Riemannian manifold $(N^{n+1},g)$. Brian White proved that this flow (defined 'weakly' as a level set flow ...
- 4,626
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Is an inner product $\langle X, \epsilon\rangle$ between log-concave $X$ and $\epsilon\gets \{0,1\}^n$ log concave?
Let $X$ be a random variable with a density $p(x)$ with respect to the Lebesgue measure. We say that $X$ is log concave if $p(x) = \exp(-V(x))dx$ for $V(x)$ a convex function.
Let $X$ be log-concave ...
- 1,602
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votes
1
answer
59
views
Are $(\Lambda,r_0)$-perimeter minimising sets $C^{1,1}$?
I've tried to find counterexamples or results in this direction, but I haven't found what I'm after (except for the $\mathbb{R}^2$ case).
Allard's regularity theorem guarantees that $(\Lambda,r_0)$-...
- 610
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votes
1
answer
220
views
A geometric criterion for uniqueness in the Plateau problem?
Let $\gamma: S^1 \to \partial B \subset \mathbf{R}^3$ be a smooth, simple closed curve in the boundary of the unit ball. Suppose that $\gamma$ intersects every horizontal plane $\Pi_t = \{ z = t\}$ at ...
- 4,626
3
votes
0
answers
80
views
A strong maximum principle for varifolds of arbitrary codimension
Let $M$ be an $n$-dimensional Riemannian manifold and $N$ a hypersurface in $M$. Let $p \in N$ and $\kappa_1 \leq \cdots \leq \kappa_{n-1}$ be the principal curvatures of $N$ at $p$ with respect to a ...
- 405
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votes
1
answer
172
views
Concentration of volume towards the boundary
Consider a Euclidean space $X$ of large dimension $N$. For a measurable subset $G\subseteq X$ and $\varepsilon>0$ let
$$G_\varepsilon:=\{x\in G\mid B_\varepsilon(x)\subseteq G\}$$
be the set of all ...
- 93
5
votes
0
answers
115
views
Regularity of the spherical mean of a compactly-supported function
The problem
Consider a $C²$ function $f: X \to \mathbb{R}$, for some compact set $X \subset \mathbb{R}^d$ with $C^1$ boundary, say $\partial X$. I am only interested in $d\in \{2,3\}$.
Then, consider ...
- 575
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votes
1
answer
132
views
What is the area-decreasing 'convex hull'?
Let $K \subset \mathbf{R}^3$ be a compact set.
What is the smallest set $C$ containing $K$, with the property that in a neighbourhood of $C$, the closest-point projection of surfaces onto $C$ ...
- 4,626
0
votes
0
answers
89
views
Finding set of best approximations from a point in $c_0$ to its subspace
Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best ...
- 19
2
votes
0
answers
121
views
Harmonic functions on varifolds
Let $T$ be a $k$-dimensional varifold in a Riemannian manifold $M$. Assume that $f$ is a smooth function on $M$ which is weakly (sub-)harmonic on $T$; that means that
$$
\int \langle \nabla_\omega f, ...
- 405
3
votes
0
answers
81
views
Are there Lojasiewicz-Simon estimates with boundary?
Let $M$ be an analytic manifold with boundary $\partial M$, equipped with a Riemannian metric $g$, which is also analytic up to and including the boundary.
Are there Lojasiewicz–Simon estimates ...
- 4,626