# 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $... 1answer 625 views ### How can we not know the$s$-measure of the Sierpiński triangle? I'm preparing a presentation that would enable high-school level students to grasp that the (self-similarity) dimension of an object needs not be an integer. The first example we look at is the ... 1answer 254 views ### Can I cover a compact set by balls {B} such that {2B} has bounded overlap? Suppose I have a compact set$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$; ... 0answers 76 views ### lower volume bound of submanifolds with small mean curvature integral data$(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$, and the ... 1answer 120 views ### Approximating measure by partitions Let$(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}$be a countable ... 1answer 128 views ### Meaning of “quantitative result” [closed] Recently I've begun reading on metric measure spaces and I keep seeing statements containing the phrase ", quantitatively". What does this mean, I googled it and couldn't find a rigorous answer. 0answers 55 views ### Arithmetic product and sum of limit sets of non-elementary Fuchsian group of second kind Let$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 \...
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 ...