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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2
votes
0answers
75 views

Is the Sobolev limit of asymptotically flat functions also flat?

Note: By a representative of a Sobolev function $f \in W^{1, p}$ I mean a genuine measurable function in the $L^p$ equivalence class of $f$. To prevent confusion, we will refer to concrete measurable ...
3
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1answer
197 views

Differential forms and continuous maps

Let $$ X \xrightarrow{f} Z \xleftarrow{g} Y $$ be smooth manifolds and smooth maps (smooth = $C^\infty$), and $$ X \xrightarrow{K} Y $$ be a continuous map such that $f=g\circ K$. Let $\...
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1answer
54 views

Interchange of integration and supremum

Let $u \in C^0(-T,T; L^2(B_R))$ be a measurable function, then is the following true? $$ \int_0^R \sup_{-T<t<T} \int_{S_r} |u(\sigma ,t)|^2 \ d \sigma \ dr = \sup_{-T<t<T}\int_0^R \int_{...
10
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0answers
312 views

Topological dimension, Hausdorff dimension, and Lipschitz mappings

I can prove the following result. Here $\operatorname{dim} X$ stands for the topological dimension and $\mathcal{H}^n$ denotes the Hausdorff measure. Theorem. Suppose that $f:\mathbb{R}^n\supset\...
1
vote
1answer
148 views

Stability of isoperimetric inequality

Let $S$ be subset of $\mathbb{R}^n$ with perimeter 1. Isoperimetric inequality states that then the volume of $S$ is not greater than $V_n$, where $V_n$ is the volume of a ball in $\mathbb{R}^n$ with ...
18
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0answers
253 views

The Dual of $BV$

I'm going to let $BV := BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the problem of characterizing $BV^*$, the dual of $BV$, which is ...
4
votes
1answer
159 views

Prove $\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx dist(x,\partial \Omega)^{-s}$, $s \in (0,2)$

Let $\Omega \subset \mathbb R^N$ and $s \in (0,2)$. Under what assumptions on $\partial \Omega$ do we have $$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx \mathrm{dist}(x,\partial \...
5
votes
1answer
122 views

Is the graph of a Sobolev function “almost geodesically complete”?

Definitions and notation: Let $\Omega$ be a open, convex, bounded subset of $\mathbb R^n$ with Lipschitz boundary, and $f \in W^{1,1}(\Omega)$ a Sobolev function. Given $x \in Ω$, we denote by $x’$ ...
7
votes
1answer
153 views

Isoperimetric type inequality in $\mathbb{R}^2$

Fix $L \in (0,\infty)$ and consider $\mathcal{C}_L$ defined as follows: \begin{align*} \mathcal{C}_L := \{ \gamma:[0,1] \rightarrow \mathbb{R}^2 |~ \gamma \text{ is smooth and length($\gamma$)$=L$ }\}....
3
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2answers
190 views

Maximal Hausdorff dimension of the set on which derivatives do not agree

Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the maximal Hausdorff dimension $d$ (and corresponding Hausdorff $d$-measure) of ...
0
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1answer
66 views

Equivalence of statements about level sets: $u|_{S \times [\tau, \infty)}$ depends only on $t$ $\iff$ $u(t,x) = \mu^{\tau}(t,u(\tau,x))$

Let $u:\mathbb R_+ \times \Omega \subset \mathbb R^N \to \mathbb R$ (sufficiently smooth). Are the following statements are equivalent? For every $\tau >0$ and level surface $S$ of $u(\tau,\cdot)$,...
3
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0answers
101 views

Volume of a set vs volume of its projections

Let $V\subset\mathbb R^n$ be "nice" (measurable or Borel or open or convex...) and let $V_{\{i,j,...\}}$ be the projection of $V$ on the subspace spanned by $e_i,e_j,...$. It is easy to see ...
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0answers
160 views

The square of a measure

Notation: We denote by $\mathcal L$ the usual Lebesgue measure on $[0, 1]$. We denote by $\mathcal P = \{a_0, ..., a_n\}$ a partition of $[0, 1]$ and $\Delta \mathcal P := \max_{0 \leq i \leq n} |a_n -...
2
votes
1answer
170 views

Does the derivative of a BV function with no jump part vanish on level sets?

Let $u: \mathbb R^n \to \mathbb R$ be a $BV$ function with no jump part, i.e., writing $Du = D^a u + D^s u + D^j u$ for the decomposition of $Du$ into absolutely continuous, Cantor, and jump part ...
5
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0answers
81 views

Minimal cones and homology spheres

Let $\Sigma \subset \mathbf{S}^{n}$ be a codimension one, embedded minimal surface in the round $n$-dimensional sphere. Let moreover $\mathbf{C} = \mathbf{C}(\Sigma)$ be the minimal cone in $\mathbf{R}...
2
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0answers
40 views

Theory of mollifiers on the boundary of a $C^2$ domain

Let $D\subseteq\mathbb{R}^d$ be a nice but not smooth domain, somewhere between Lipschitz and $C^2$. I am looking for a reference on the theory of mollifiers and regularization for functions on $\...
4
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0answers
233 views

Investigating a 1/2 -dimensional sphere and defining a fractional Euclidean space

A small note I'm a new member on MO and I'm not sure if this question fits in here. If not, please don't be too hard on me. I am transferring a question I asked on MSE in here, because the user who ...
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0answers
70 views

Lebesgue measure of a neighbourhood of a curve

Let $\Omega\subseteq\mathbb{R}^N$ be an open, bounded and with smooth boundary (e.g. Lipschitz boundary or more if necessary). For any function $\phi:\Omega\to\mathbb{R},\ \phi\in C^1(\overline{\Omega}...
0
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1answer
140 views

Lebesgue measure of sets in $\mathbb{R}^N$

Let $\Omega\subseteq \mathbb{R}^N$ be an open, bounded and connected set (it can be assumed with smooth boundary if necessary). Consider $\phi:\Omega\to\mathbb{R}$, $\phi\in C^1(\overline{\Omega})$ (...
2
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0answers
106 views

Metric on space of Borel-measurable functions

Let $(X,d_X),(Y,d_Y)$ be metric spaces and $X$ is locally-compact and fix a Borel probability measure $\nu$ on $X$. For any Borel-measurable $f:X\rightarrow Y$, let $\mathcal{K}(f,\delta)$ be the set ...
5
votes
1answer
160 views

Proof of Denjoy-Riesz Theorem and Moore's Generalization?

The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane can be covered by an arc, i.e. an embedded image of $[0,1]$. Sometimes it's stated just for covering a Cantor Set,...
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0answers
44 views

Sequence of open sets converge in characteristic function to an open set?

Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with Lipschitz boundary. Consider a sequence of open sets $\omega_n\subseteq\Omega,\ n\in\mathbb{N}^*$ such that there is a Lebesgue ...
2
votes
1answer
123 views

Hausdorff dimension and surface measure

Could someone please indicate me some reference that contains the proof of the following theorem? Below $\mathcal{H}^n$ denotes the $n$-dimensional Hausdorff outer measure in $\mathbb{R}^n$. Theorem: ...
3
votes
0answers
65 views

Which stationary varifolds have non-integer density?

A central object in geometric measure theory are the generalised, and weakly defined minimal surfaces called stationary varifolds. Let me recall some definitions. Given an open subset $U \subset \...
3
votes
0answers
86 views

Pushforward of measures with Fourier decay

Suppose $\gamma: [0,1]^d \to \mathbf{R}^{d+1}$ is a smooth map with nonvanishing Gaussian curvature, and $\mu$ is a probability measure compactly supported on $(0,1)^d$ such that $|\widehat{\mu}(\xi)| ...
8
votes
1answer
517 views

How to shrink a square with minimal distortion?

$\newcommand{\CO}{\text{CO}_2}$ $\newcommand{\euc}{\mathfrak{e}}$ $\newcommand{\SO}{\text{SO}_2}$ $\newcommand{\al}{\alpha}$ $\newcommand{\dist}{\text{dist}}$ $\newcommand{\Lip}{\text{Lip}_{\text{inj}}...
3
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0answers
70 views

Isotopy Classes and Embeddability of Products in $\mathbb{R}^2$

On MSE I asked if the plane contains an uncountable collection of mutually disjoint copies of the Warsaw Circle; it seems to be false, and is probably already known but I'm not sure that anybody has ...
4
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0answers
223 views

Are BV functions “almost continuous”?

Let $\Omega$ be the open cube $(0, 1)^n$, $n \geq 2$, and $\mu$ the Lebesgue measure. Denote by $A$ the set of Lebesgue measurable subsets of $\Omega$ with measure $1$. For any $f \in L^\infty (\Omega)...
0
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1answer
77 views

Estimate on total variation of composition of functions

Let $f \in BV(\mathbb R)$ and $g: \mathbb R \to \mathbb R$ be Lipschitz. How can I estimate the total variation of $f\circ g$, that is $$ \int_{\mathbb R} \left|\frac{d}{dx}f(g(x))\right| dx \ ? $$ ...
3
votes
1answer
142 views

Average of the sum of dirac measures

Let $(M^n,g)$ be a closed smooth Riemannian manifold. Consider a set $\mathcal B_{\epsilon}$ which consists of a maximal number of points in $M$ with pairwise distance no smaller than $\epsilon$. We ...
1
vote
0answers
78 views

On the uniform boundedness principle and the space of functions of bounded variation

Let $U$ be a bounded smooth domain of $\mathbb{R}^d$. We write $m$ for the Lebesgue measure on $U$. A function $f \in L^1(U,m)$ has bounded variation in $U$ if \begin{align*} V(f,U):=\sup \left\{\int_{...
0
votes
0answers
68 views

The volume of boundary layer

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set with $C^2$ boundary $\partial\Omega$. Let $\operatorname{d}(x):=\inf_{y\in\partial\Omega}|x-y|$ for $x\in\overline{\Omega}$, and the open set $\...
1
vote
1answer
159 views

Relationship between volume density and area density

Let $\mu(x)dx$ be a measure in $\mathbb{R}^{2n-2}$, where $\mu$ (a $C^\infty$ and positive function) is the density of the volume in the sense that $\DeclareMathOperator{\Vol}{\mathrm{Vol}} \Vol_\mu(...
1
vote
0answers
131 views

Does a sequence of Jacobians converge to the 'correct' continuous part plus some controlled singular part?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \in W^{1,...
0
votes
0answers
43 views

Examples of strongly continuous measure-valued functions

Let $X$ be a compact geodesic metric space and let $P_p(X)$ be the set of all finite Borel measure on X with finite $p^{th}$ moment. We equip $P_p(X)$ with the total variation topology metric. What ...
1
vote
1answer
165 views

Does weak continuity of Jacobians hold for non nondegenerate maps?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \...
2
votes
0answers
145 views

Two generalizations of the Verblunsky Theorem

I learned from this paper about the Verblunsky theorem. My question is that: What kind of generalizations of this theorem is availlable? In particular I am interested in the following two possible ...
1
vote
1answer
205 views

Weak continuity of law

Let $\mathcal{P}_2(\mathbb{R}^n)$ denote the set of all Borel probability measures on $\mathbb{R}^n$ with finite variance and weak topology. Let $X_t$ be a strong solution to the SDE with initial ...
1
vote
1answer
98 views

A Frostman-type result for measures satisfying uniform lower density conditions

Let $\mu$ be a finite, compactly supported, non-zero measure on $\mathbb{R}^d$ for an integer $d$. Let $B(x,r)$ denote the ball of radius $r>0$ centered at $x \in \mathbb{R}^d$. For $\delta \in [0,...
8
votes
2answers
203 views

Averaging the mass of a Sobolev function $f\in W^{1,p}(\Omega)$ near $\partial\Omega$

Recently, I asked a somewhat related question here. In the comment section, I found the formula $$ \lim_{r\to 0}\frac{1}{r}\int_{\Omega_r} f(x)\,dx = \int_{\partial \Omega}f(\sigma)\,d\mathcal{H}^{n-1}...
0
votes
0answers
100 views

Barycenters on Hadamard Manifolds

Let $(M,g,m_0)$ be a pointed-Hadamard manifold with Riemmanian distance function $d_g$, $(X,\Sigma,\mu)$ be a finite measure space. We use $L^2(\mu;M,m_0)$ to denote the metric space consisting of ...
4
votes
0answers
115 views

Estimating the size of $\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}$

Let $\Omega$ be a bounded domain in $\Bbb R^n$. Define $$ \Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}, $$ i.e. it the ring of thickness $r$ at the boundary of $\Omega$. Intuitively, ...
2
votes
0answers
117 views

Equality of Hausdorff measure and Lebesgue measure on manifolds (reference)

Let $\mathcal{M} \subset \mathbb{R}^N$ be an $n$-dimensional $C^1$ submanifold (connected). We have two metric functions on $\mathcal{M}$: The Euclidean distance inherited from $\mathbb{R}^N$. The ...
2
votes
0answers
227 views

Can be this “handwaving” idea about “counting” reals somehow put on solid ground?

We know that the Cantor's cardinality of a countable set is $\aleph_0$ and the cardinality of continuum is $2^{\aleph_0}=\aleph_0^{\aleph_0}$. Unfortunately, this measure is based on the idea of ...
1
vote
1answer
234 views

Why is the Hausdorff measure of this set zero?

Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set, and let $\phi:\Omega\to\mathbb{R}^N$ be a $C^1$ function with the property that $\phi^{-1}(0)\neq\emptyset$, and $\nabla\phi(x)\neq 0,\ \...
5
votes
1answer
228 views

Signed distance function and level set

For $\phi\in C^1(\mathbb{R}^N)$ with $$\omega_{\phi}=\{x\in\mathbb{R}^N\ |\ \phi(x)>0\}$$ being a bounded set with $\nabla\phi (x)\neq 0,\ \forall\ x\in\phi^{-1}(0)=\partial\omega_{\phi}\neq \...
0
votes
0answers
69 views

Signed distance function

Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with uniform Lipschitz boundary. Consider the signed distance function: $d:\mathbb{R}^N\to\mathbb{R},\ d(x)=\begin{cases} \mathrm{dist}(x,\...
5
votes
2answers
168 views

Comparison of Information and Wasserstein Topologies

There are many possible metrics one can place on the space of Gaussian probability measures on $\mathbb{R}^n$, with strictly positive definite co-variance matrices. Let's denote this space by $X$. I'...
6
votes
3answers
458 views

How to estimate the integral involving the distance function

Let $\Omega\subset\mathbb{R}^n$ be an open bounded domain with smooth boundary. Consider the following integral: $$I(t)=\int_{\Omega}e^{-\frac{d^2(y,\partial\Omega)}{t}}{\rm d}y.$$ My problem is how ...
2
votes
0answers
98 views

Extensions of minimal hypersurfaces

Let $B \subset \mathbf{R}^{n+1}$ be the unit ball, and $M \subset B$ be a minimal hypersurface. By this we mean that $M$ is an embedded $n$-dimensional submanifold with vanishing mean curvature. We ...

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