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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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 \...
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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
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1answer
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
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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
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2answers
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 ...
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1answer
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 ...
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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
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3answers
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
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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 \...
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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 ...
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1answer
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) &= \...
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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 $T=i^{(n-p)^{2}} \sum T_{I, J} d z_{I} \wedge d \bar{z}_{J}$ ...
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2answers
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
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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 ...
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0answers
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\...
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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 "...
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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
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1answer
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 ...
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0answers
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\...
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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
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1answer
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 \...
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0answers
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 $\mathbb{R}^d$ ...
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0answers
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 ...
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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 ...
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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
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1answer
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 ...
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1answer
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 $$ \...
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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 ...
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1answer
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 ...
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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
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0answers
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 ...
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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 ...
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1answer
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?
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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
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2answers
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
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1answer
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
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3answers
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
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1answer
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
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1answer
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 \...
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0answers
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{...
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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
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1answer
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
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1answer
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 $j$; ...
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0answers
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
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1answer
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 ...
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1answer
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.
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0answers
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
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2answers
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 ...
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0answers
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(...
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0answers
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_{\...

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