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.
763 questions
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429
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Two equivalent measures on the real Grassmannians?
Denote by $G(n,k)$ the real Grassmannian, the set of $k$-dimensional subspaces of $\mathbb{R}^n$. It is a topological space, even metrizable (see A metric for Grassmannians), and so it is a ...
- 859
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0
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194
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Class of flat currents stable under $\overline{\partial}$ operator
Given $U\subset\mathbb{C}^n$, open domain, a locally flat current on $U$ is a $k-$current $T$ such that for every $f\in\mathcal{D}(U)$ (smooth functions with compact support in $U$) there exist a ...
- 1,205
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334
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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(...
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2
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184
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Intersection with a fixed set in Hausdorff metric space [closed]
Let $\mathbb{R}^d$ be a the usual Euclidean space and let $Y$ be a fixed non-empty closed subset of $Ball(0,1)$ (the unit ball in $\mathbb{R}^d$ about $0$ of radius $1$).
Let $f$ be the map taking $...
- 5,405
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1
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541
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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?
$$
\int_0^R \sup_{-T<t<T} \int_{S_r} |u(\sigma ,t)|^2 \ d \sigma \ dr = \sup_{-T<t<T}\int_0^R \int_{...
- 455
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1
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99
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Is Fourier dimension finitely stable?
Let $A,B$ be compact subsets of $\mathbb R$. Let $a=\mathrm{dim}_F(A)$, $b=\mathrm{dim}_F(B)$ be their Fourier dimensions, respectively. My questions are as follows:
Is it true that $\mathrm{dim}_F(A\...
- 751
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67
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Hahn-Mazurkiewicz with finite one-dimensional Hausdorff measure
Suppose that there is a continuous surjection from $[0,1]$ to a metric space $(X,d)$. If $(X,d)$ has finite one-dimensional Hausdorff measure, must there exist a Lipschitz surjection from $[0,1]$ to $(...
- 133
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2
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308
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Calculating the Fourier dimension of a real interval $\left[a, b\right]$
(Preliminaries:) 1.) Let $S\subset\mathbb{R}^n$ and define $\mathcal{M}(S) = \{\text{$\mu$ a Borel measure}: \text{$0 < \mu(S) < \infty$ and $\mathrm{support}(\mu)\subset S$}\}$.
2.) Define the ...
- 151
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232
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When does $C_b(X)$ admit a Schauder Basis?
Let $(X,d)$ be a separable and connected metric space. My question is rather short and to the point: do there exist $\{x_n\}_{n=0}^{\infty}\subseteq X$ such that
$$
\left\{d(x_n,\cdot)-d(x_0,\cdot)\...
- 195
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227
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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 $\phi:\Omega\to\mathbb{R}$, $\phi\in C^1(\overline{\Omega})$ (...
- 1,759
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75
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Fractal set $E$ such that the indicator function $\mathbf{1}_E$ is BV
Is there a "fractal" set $E \subset \mathbb R^2$ such that the indicator function $\mathbf{1}_E$ is in $BV(\mathbb R^2)$?
- 839
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298
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Hausdorff measure of the zero set
Let $f : \mathbb R^n\to \mathbb R$ continuous, for which there exist $x,y\in\mathbb R^n$, such that $f(x)f(y)<0$.
Is it true that the Hausdorff dimension of the zero set of $f$ is at least $n-1$?
- 2,933
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If $ \mathcal{H}^k(B_1(0)\cap S)\leq A\omega_k $ when $ \mathcal{H}^k(B_r(x)\cap S)\leq A\omega_kr^k $ for all $ 0<r<\delta $, $ x\in\mathbb{R}^n $?
Let $ S\subset\mathbb{R}^n $ is of finite $ k $-dimensional Hausdorff and $ 0<\delta<1 $ is a constant. If for any $ x\in\mathbb{R}^n $ and $ r>0 $, we hae
$$
\mathcal{H}^k(S\cap B_r(x))\leq ...
- 1,219
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72
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Equivalence of statements about level sets: $u|_{S \times [\tau, \infty)}$ depends only on $t$ $\iff$ $u(t,x) = \mu^{\tau}(t,u(\tau,x))$
Let $u:\mathbb R_+ \times \Omega \subset \mathbb R^N \to \mathbb R$ (sufficiently smooth). Are the following statements are equivalent?
For every $\tau >0$ and level surface $S$ of $u(\tau,\cdot)$,...
- 839
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1
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236
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Estimate on total variation of composition of functions
Let $f \in BV(\mathbb R)$ and $g: \mathbb R \to \mathbb R$ be Lipschitz. How can I estimate the total variation of $f\circ g$, that is
$$
\int_{\mathbb R} \left|\frac{d}{dx}f(g(x))\right| dx \ ?
$$
...
- 303
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1
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132
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Projection onto manifold of Gaussian measures by "trunction" of moments
Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^n$ with finite mean and variance; in the sense that
$$
\int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \...
- 5,405
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1
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189
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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?
- 109
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1
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182
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Does every compact doubling metric space have a canonical measure?
My question is this one, with the additional condition that the metric space be doubling. In the aforementioned question, the limiting measure depends on the sequence $\epsilon_n$ and hence is not ...
- 6,541
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127
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Boundary values of $f$, bounded linear operator
I have a question about Sobolev spaces
Let $U$ be a bounded Lipschitz domain of $\mathbb{R}^{d}$. $H^{1}(U)$ denotes the first order $L^2$-Sobolev space on $U$ with Neumann boundary condition.
It ...
- 721
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1
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297
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Approximating characteristic functions by cutting the real axis into smaller and smaller pieces
Let $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z} \subset\mathbb{R} (r>0)$, let $E\subset\mathbb{R}$ be a Lebesgue measurable set with finite measure $|E|$, define $J_r=(-\frac{1}{4\pi r}, \frac{1}{4\pi ...
- 1,906
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168
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The partition of $N-1$ rectifiable set
The updated version can be found here.
Let $S\subset \mathbb R^N$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<\infty$.
My question, for each $x_0\in S$, would it be possible to choose a ...
- 679
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628
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Lipschitz boundary vs rectifiable curve boundary
I was looking at an old paper about domains with Lipschitz boundary. I am wondering, suppose that the boundary of a compact domain homeomorphic to a disk is a rectifiable injective curve : is this ...
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549
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One-dimensional Hausdorff measure of preimages
Let $\Omega$ be an open subset of $\mathbf{R}^n$. For a mapping $f: \Omega\to \bf{R}^n$, what kind of condition ensures that the one-dimensional Hausdorff measure of $f^{-1}(E)$ is zero whenever $E$ ...
- 1,881
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154
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optimize with respect to domain shape
Let $\Gamma$ be the set of all closed $C^2$ curves in the plane which enclose unit area and let $\Omega$ be the set of all subsets of $\mathbb{R}^2$ that are enclosed by some curve in $\Gamma$. Now ...
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A general rule for approximating the perimeter of a set with finite perimeter in terms of the volume
I want to know if it is possible to have a general rule for approximating the perimeter of a set $E\subset \mathbb{R}^n$ with finite perimeter in terms of the volume (Lebesgue measure) of a sequence ...
- 747
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77
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Wasserstein space isomorphic to original space?
Is there a complete measurable metric space $(X,d)$ for which its $p$-Wasserstein space $W(X)$ is isometrically isomorphic to $(X,d)$ for some $p \in [1,\infty]$?
Note that there is a canonical non-...
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36
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On the Lipschitz parametrizability of polynomials of fixed Mahler measure
Background
For a polynomial $f(x) = a(x-\alpha_1) \cdots (x - \alpha_n) \in \mathbb{C}[x]$, its Mahler measure is defined to be
$$M(f) = |a| \prod_{i=1}^n \max\{1, |\alpha_i|\}$$
In Lemma 1, Masser-...
- 267
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Why does $\omega$ belong to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-2 + n\varepsilon^3}$ different $\theta$?
In this paper, there is the following claim (Pg. 1850):
If $1 - \eta(w) \ne 0$, then $|\omega| \ge \rho$. In that case, $\omega$ belongs to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-...
- 165
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94
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Bounding the area of the image of a set by product of maximum of lengths
Let $F:[0,1]\times[0,1]\to \mathbb {R}^2$ be a smooth function. Given $x\in [0,1]$, let $\ell_x:=\{x\}\times [0,1]$, and given $y\in [0,1]$, let $\ell_y:=[0,1]\times \{y\}$.
My question feels ...
- 502
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94
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When can an affine functional on the dual be represented as an element of a Banach space?
In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
- 297
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410
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Properties of doubling metric spaces
At present I work with tools that involves doubling metric space, my definition of DME is:
A metric space $X$ is called doubling with constant $N$, where $N \geq 1$ is an integer, if, for each ball $...
- 101
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87
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How to find a smallest parallelepiped that bounds the unit ball in a normed space
Consider a finite dimensional normed space $(V,\Vert \cdot \Vert)$. How to find a basis $(e_i)$ of $V$ such that the unit closed ball $\overline B_1$ centered at $0$ is contained in $ P:= \{ x \in V : ...
- 423
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92
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Finding set of best approximations from a point in $c_0$ to its subspace
Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best ...
- 85
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Distortion estimates to control Hausdorff measure of a curve
I am studying the paper Blumenthal - Statistical properties for compositions of standard maps with increasing coefficent.
I have a problem to understand how the distortion estimates are used. The ...
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43
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Minimal condition on set for an optimisation problem
We fix $\Omega \subset \mathbb{R}^{2}$ an open set. My question is: what are the minimum conditions we need on $E \subset \Omega$ such that the following optimisation problem:
$$
\sup\{ \int_{E}(\...
- 181
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70
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Measure and other properties of nodal domains of Laplacian
Let $(\phi_k,\lambda_k)$ be the couple of eigenfunctions and eigenvalues of the the Laplacian operator on $\Omega \subset \mathbb R^n$.
The nodal set of $\phi_k$ is the set $$\mathcal N_k = \{x \in \...
- 161
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57
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Can iterative application of ham sandwich cuts form streamlines of an ODE?
It has been known that given two probability distributions $\mu_1$ and $\mu_2$ (let us say, they are smooth for simplicity), there is a hyperplane that divides the domain into two regions (denoted as $...
- 23
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425
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Compact connected Riemannian manifolds are Ahlfors regular metric space
Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,...
- 5,405
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69
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Holder-continuous barycenter maps
Let $(X,d)$ be a complete locally-compact metric space. We define the $p$-barycenter map as a continuous function:
$$
\beta:\mathcal{P}_p(X)\rightarrow X,
$$
which is a right-inverse of the map ...
- 5,405
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235
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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}...
- 1,759
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0
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88
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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 ...
- 1,759
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76
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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 $\...
- 71
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59
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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 ...
- 5,405
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0
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131
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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 ...
0
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0
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216
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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,\...
- 1,759
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0
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96
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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 $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
- 167
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66
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Is there a dyadic cube decomposition where edge length is comparable to L^2 averages?
Suppose I have some measurable function $f : B_1(0) \to [0,1)$, which is pointwise very small, i.e. $\|f\|_{\infty} << 1$.
I'm looking to construct some kind of dyadic cube decomposition or ...
- 1,179
0
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0
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95
views
Hausdorff dimension of level set of Conway base 13 function
Yesterday I had a discussion about Conway's base-13 (https://en.wikipedia.org/wiki/Conway_base_13_function) function (and what fancy properties it has). During that discussion the other person ...
- 1,045
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0
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63
views
Coarea-like formula for BV functions (not their derivative)
Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that
$$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$
Unfortunately, the formula
$$f = \int_{\mathbb R} \...
- 839
0
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0
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156
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Function classes with high Rademacher complexity
My question is two fold,
Is there any general understanding of what makes a function class have high Rademacher complexity? (Sudakov minoration would say that one sufficient condition for a class of ...
- 2,246