# 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.

440
questions

**2**

votes

**0**answers

50 views

### Is the set $\operatorname{Unif}(0,\frac{1}{n})$ for odd and even $n$ a 2-alternating capacity?

Let $\Omega$ be a complete metrizable space $\mathscr A$ its Borel $\sigma$-algebra and $\mathscr M$ the set of all probability measures on $\Omega.$ Every non-empty subset $\mathscr P \subset \...

**2**

votes

**0**answers

51 views

### Lebesgue measure of set of equidistant points with respect to a finite set

Let $X$ be a finite subset of $\mathbb{R}^n$; equip $d$ with a metric, and let $\emptyset \subset X\subseteq \mathbb{R}^n$ be of cardinality $N>0$.
What requirements on my metric do I need so ...

**4**

votes

**1**answer

77 views

### Domains in $\mathbb{R}^n$ for which Hajlasz-Sobolev spaces and Sobolev Spaces are the same

I'm reading Heinonen's book on metric measure spaces. He writes that for general domains $\Omega \subset \mathbb{R}^n$, $M^{1,p}(\Omega) \subset W^{1,p}(\Omega)$ where the former are Hajlasz-Sobolev ...

**4**

votes

**0**answers

107 views

### Continuous disintegration

Given a suitable Borel measure $\mu$ on a suitable topological space $X$ and a Borel function $\pi:X \to Y$, where $Y$ is another suitable topological space, the disintegration theorem gives a Borel ...

**6**

votes

**2**answers

340 views

### Unknown work of Nöbeling on topological/Hausdorff dimension

Let $\mathcal{H}^n$ denote the Hausdorff measure, $\dim_H X$ the Hausdorff dimension, and $\dim X$ the topological dimension of $X$.
A well known result of
Szpilrajn (He changed his name to ...

**-1**

votes

**1**answer

77 views

### Isometric stratification preserves volume?

Let $K\subset \mathbb{R}^k$ be a non-empty compact subset let $f:K \to K$ be Lipschitz and surjective. If, moreover, $f$ is an isometry then clearly $f$ preserves the Lebesgue measure of $K$.
I ...

**1**

vote

**0**answers

53 views

### zero extension of positive currents are always positive

In Demailly's Complex Analytic and Differential Geometry page 139:
He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...

**2**

votes

**3**answers

142 views

### Covering families of sets by small-measure partitions

Let $(X,\mathscr{A},\mu)$ be a probability space and let $\{A_1,\ldots,\}\subset\mathscr{A}$ be a countable family of sets with small measure: say $\mu(A_i)\le\epsilon$. I am trying to show that one ...

**3**

votes

**0**answers

141 views

### Sets of finite perimeter: intersection with an half space

I have a question regarding sets of finite perimeter. In particular I'm interested to find
$$\mu_{E \cap H_t}, \label{1}\tag{1}$$
where $E$ is a set of finite perimeter in a generic open set $\Omega \...

**2**

votes

**0**answers

51 views

### Closure theorem for weak limits of “foliation currents”

A "foliation current" in the sense of Ruelle-Sullivan (https://www.math.stonybrook.edu/~ebedford/PapersForM655/RS.pdf) is essentially a closed subset of a manifold foliated by equidimensional oriented ...

**1**

vote

**1**answer

70 views

### Convergence of probability measures which (asymptotically) concentrate along a submanifold

Let $V : (-1, 1)^d \to \mathbf{R}_+$ be a smooth function, and for $\beta > 0$, define
\begin{align}
P_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\
z (\beta) &= \...

**2**

votes

**0**answers

50 views

### About the current of finite mass

In Demailly's e-book Complex analytic and differential geometry,
chap3-(1.14) Proposition is stated as follows:
Every positive current $T=i^{(n-p)^{2}} \sum T_{I, J} d z_{I} \wedge d \bar{z}_{J}$ ...

**8**

votes

**2**answers

226 views

### Graph metric approximating Euclidean metric

I've been reading Wolfram's recent articles about graph/mesh/grid structures as an analogy for physical space, and it seems to me that there will be a problem getting the notion of distance to work ...

**4**

votes

**0**answers

154 views

### Classification of Euclidean-invariant measures?

Is there a classification of measures on $\mathbb R^n$ which are invariant under (Euclidean) isometries? Hausdorff measures of all kinds are examples -- could that be all of them? More precisely,
By ...

**1**

vote

**0**answers

46 views

### Compatibility of the Hausdorff measure with short exact sequences in normed spaces

Let $(E,\|.\|)$ be a finite dimensional normed space and take $F\subset E$ a
subpace, so that we have the canonical short exact sequence
$0\rightarrow F\rightarrow^\iota E\rightarrow^\pi E/F\...

**2**

votes

**0**answers

88 views

### Is Steiner symmetrization “Turing complete”?

This question stems from intuition so it is a little soft. It concerns performing computation using transformations on sets. The idea is that a rearrangement like Steiner symmetrization might be "...

**1**

vote

**0**answers

49 views

### Continuity haussdorff measure w.r.t level set and coarea formula

Let $M$ be a smooth compact riemannian manifold of dimension $n$ without boundary, and $A$ a measurable subset of $M$. Let $f:M\mapsto\mathbb{R}$ be a $C^1$ function and $\varphi:\mathbb{R}\mapsto\...

**4**

votes

**1**answer

108 views

### Every convex set is of locally finite perimeter

I need to prove that every convex subset of $\mathbb{R}^n$ is of locally finite perimeter.
$E$ is of locally finite perimeter if there exists a vector-valued Radon measure $\mu_E$ s.t. the Gauss ...

**2**

votes

**0**answers

49 views

### For $\mathcal{L}^1$-a.e. $t\in R$, Hausdorff dimension of level sets of a locally Lipschitz function $f:R^n\to R$ is $n-1$?

Let $f:R^n\to R$ be a locally Lipchitz function. Denote $H^n$ the n-dimensional Hausdorff measure. We know that for any $H^n$-measurable subset $A\subset R^n$, for $\mathcal{L}^1$-a.e. $t\in R$, $A\...

**2**

votes

**0**answers

67 views

### First Dirichlet eigenvalue below second Neumann eigenvalue?

Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary.
I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...

**2**

votes

**1**answer

209 views

### Bounding an “integral” from below by the Hausdorff measure of the domain

Let $(X,d)$ be an arbitrary metric space and $E \subset X$ also arbitrary. Fix $s \in (0,\infty)$.
Is it true that for any $ \delta > 0 $ and any collection of pairs $\{(A_i,a_i)\}_{i \in \...

**1**

vote

**0**answers

97 views

### Reference request: harmonic analysis with non-Lebesgue reference measure

The Lebesgue measure on $\mathbb{R}^d$
admits the following polar decomposition:
$$
L(dx) = r^{d-1} dr \lambda(dy),
$$
where 𝜆 is the uniform measure on the Euclidean unit sphere of $\mathbb{R}^d$ ...

**1**

vote

**0**answers

36 views

### Decomposition of the space of Radon measures with respect fractional harmonic capacity?

It is well know that there is a generalization of Lebesgue decomposition theorem in the following way:
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...

**4**

votes

**0**answers

126 views

### Is the rectifiability of currents independent of the choice of Riemannian metric?

I apologize if this is a trivial question – GMT is not my area of expertise but I'm working my way through a proof that makes extensive use of GMT and I haven't been able to find an answer to my ...

**5**

votes

**0**answers

62 views

### Minimizing area in relative homology class

A well known result in geometric measure theory asserts that if $(M^{n+1}, g)$ is a closed Riemannian manifold and $\alpha \in H_n(M)$ is a nonzero homology class, then there exists a closed embedded ...

**2**

votes

**1**answer

84 views

### Generalization of approximate tangent spaces to subsets of arbitrary manifolds?

I'm anything but an expert in Geometric Measure Theory, so please forgive me if I'm asking a trivial question.
Let $(M^n, g)$ be a smooth Riemannian manifold, $d \in \mathbb{N}$ and $A ⊂ M$ be a ...

**1**

vote

**1**answer

123 views

### How to prove space of non-negative Radon measures is complete?

Let $\mathcal{M}^{+}(\mathbb{R}_{+})$ be space of non-negative Radon measures on $\mathbb{R}_{+}$ with bounded total variation and define the metric $\rho$ on $\mathcal{M}^{+} (\mathbb{R}_{+})$ as $$ \...

**9**

votes

**0**answers

228 views

### Second order differentiability of convex functions

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is ...

**3**

votes

**1**answer

99 views

### Generalized Nikodym sets

Let me quote en.wikipedia about the original Nikodym's example (the definitions above I've written myself just for MO):
a Nikodym set is a subset of the unit square in $\ \mathbb R ^2\ $ with the ...

**2**

votes

**0**answers

137 views

### Fubini's theorem on arbitrary foliations

In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...

**2**

votes

**0**answers

95 views

### Caratheodory's theorem in any compact Riemann surface

The classical Caratheodory theorem states that if $G$ is a simply connected domain in the plane, whose boundary is a Jordan curve, then the Riemann uniformization extends continuously to a ...

**0**

votes

**0**answers

61 views

### If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?

Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...

**0**

votes

**1**answer

92 views

### Visualization of the disintegration theorem [closed]

Where can I find a picture that gives a visualization of the disintegration theorem?
If such reference does not exist, what would a nice visualization of this fundamental result look like?

**0**

votes

**0**answers

75 views

### Area of Disc that Intersects Another under Smooth Flow

The following question can be asked in any $\mathbb{R}^n$ for $n > 1$, but the case of interest is (thankfully) the case $n = 2$. The formulation of the problem with discs isn't actually critical ...

**6**

votes

**2**answers

158 views

### Continuous section of support - Is it possible to map compact sets to measures supported on them?

Let $(X,d)$ be a compact metric space and let $(\mathcal K(X),d_H)$ and $(\mathcal P(X),d_W)$ denote its space of nonempty compact subsets with Hausdorff metric $d_H$, and its space of Borel ...

**2**

votes

**1**answer

111 views

### Measure of random Voronoi cell

Let $\mu$ be some distribution (with density) on $\mathbb{R}^d$, from which we independently draw $X_1,\ldots,X_n$. These induce a Voronoi partition on $\mathbb{R}^d$: $V_1$ is the set of all points ...

**2**

votes

**3**answers

321 views

### A Curved/Warped Version of Fubini's Theorem

I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...

**6**

votes

**1**answer

459 views

### Possible application of divergence Theorem?

suppose that $f \in C^1 (\mathbb{R}^{N+1},\mathbb{R})$. It's well known that if all his points are regular points i.e.
$$\nabla f (x) \neq 0 \; \; \; \forall x \in \mathbb{R}^{N+1}$$
then, for every ...

**6**

votes

**1**answer

168 views

### Is Hausdorff Measure equal to Hausdorff Content on rectifiable (metric) spaces?

Let $(X,d)$ be an $\mathcal{H}^n$-rectifiable metric space, i.e. there exits a collection of Lipschitz maps from measurable subsets of $\mathbb{R}^n$ to $X$ such that $ \mathcal{H}^n(X \backslash \...

**5**

votes

**0**answers

81 views

### How do sets with unit fractional Hausdorff measure of dimension $>1$ look like?

Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask:
Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{...

**1**

vote

**0**answers

62 views

### Obstacle problems for minimal hypersurfaces

Given a compact Riemannian $n+1$-manifold $M$ with (possibly not mean convex) boundary (smooth or probably with codim $>2$ singularities). Consider the following problems,
1) fix a homology class $...

**22**

votes

**1**answer

625 views

### How can we not know the $s$-measure of the Sierpiński triangle?

I'm preparing a presentation that would enable high-school level students to grasp that the (self-similarity) dimension of an object needs not be an integer. The first example we look at is the ...

**5**

votes

**1**answer

254 views

### Can I cover a compact set by balls {B} such that {2B} has bounded overlap?

Suppose I have a compact set $K \subset B_1(0) \subset \mathbb{R}^n$. Can I always find a family of open balls $\{B_{r_j}(x_j)\}$ such that
$x_j \in K$ and $B_{r_j}(x_j) \subset B_1(0)$ for each $j$;
...

**2**

votes

**0**answers

76 views

### lower volume bound of submanifolds with small mean curvature integral data

$(M^n,g)$ is a smooth submanifold in $\mathbb{R}^p$ ,and $B_1$ is the unit ball centered in the origin 0.
Is there a $\epsilon >0$，
when assuming $\int_{M\cap B_1} |H|^n \leq \epsilon$, and the ...

**1**

vote

**1**answer

120 views

### Approximating measure by partitions

Let $(X,\mathcal{B},\mu)$ be a non-atomic Borel probability space. We may assume that $X\subseteq \mathbb{R}^d$ is the open (or closed) unit ball, if it helps.
Let $\mathcal{C}$ be a countable ...

**1**

vote

**1**answer

128 views

### Meaning of “quantitative result” [closed]

Recently I've begun reading on metric measure spaces and I keep seeing statements containing the phrase ", quantitatively". What does this mean, I googled it and couldn't find a rigorous answer.

**1**

vote

**0**answers

55 views

### Arithmetic product and sum of limit sets of non-elementary Fuchsian group of second kind

Let $L \subset \mathbb{R}$ be a limit set of a Fuchsian group $\Gamma$. If $\Gamma$ is a non-elementary Fuchsian group of second kind, then $L$ is a Cantor set. For example: $\Gamma= \bigg\langle \...

**5**

votes

**2**answers

288 views

### Concrete description of lift in Arens-Eells space

Let $X$ be a compact pointed metric subspace of the $d$-dimensional Euclidean space $(\mathbb{R}^d,d_E)$ and let $AE(X)$ denote its Arens-Eells space. Then a result of Nik Weaver shows that for every ...

**0**

votes

**0**answers

53 views

### Definition of perimeter question

It is well known the formula for perimeter of a set $\Omega\subset\mathbb{R}^2$, in term of the Heaviside function $H$ and Dirac $\delta$:
$$
\int_{\partial\Omega} 1\ d\sigma=\int_{\Omega} ||\nabla H(...

**1**

vote

**0**answers

90 views

### Total Mean Curvature as a integral on the whole space

It is well known from De Giorgi that we may express the surface area of a domain $\Omega\subset\mathbb{R}^N$ as:
$$
\int_{\partial\Omega} 1\ d\sigma=\int_{\Omega} ||\nabla H(\phi(x))||\ dx=\int_{\...