# 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.

527
questions

**2**

votes

**0**answers

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**

votes

**1**answer

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 $\...

**0**

votes

**1**answer

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**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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

**1**answer

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,...

**0**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**3**answers

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

**0**answers

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 ...