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.
287 questions with no upvoted or accepted answers
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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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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 ...
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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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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-...
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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-...
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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 ...
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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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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 $...
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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 : ...
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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 ...
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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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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:
$$
\sup\{ \int_{E}(\...
- 181
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Measure and other properties of nodal domains of Laplacian
Let $(\phi_k,\lambda_k)$ be the couple of eigenfunctions and eigenvalues of the the Laplacian operator on $\Omega \subset \mathbb R^n$.
The nodal set of $\phi_k$ is the set $$\mathcal N_k = \{x \in \...
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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 $...
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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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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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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}...
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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 ...
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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 $\...
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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 ...
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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 ...
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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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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 $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
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Is there a dyadic cube decomposition where edge length is comparable to L^2 averages?
Suppose I have some measurable function $f : B_1(0) \to [0,1)$, which is pointwise very small, i.e. $\|f\|_{\infty} << 1$.
I'm looking to construct some kind of dyadic cube decomposition or ...
- 1,179
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Hausdorff dimension of level set of Conway base 13 function
Yesterday I had a discussion about Conway's base-13 (https://en.wikipedia.org/wiki/Conway_base_13_function) function (and what fancy properties it has). During that discussion the other person ...
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Coarea-like formula for BV functions (not their derivative)
Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that
$$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$
Unfortunately, the formula
$$f = \int_{\mathbb R} \...
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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 ...
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Absolute continuity of probability measures determined by dependence structure
We are on $\mathbb{R}^d$ with Borel $\sigma$-algebra. Let $\mu_1, ..., \mu_d$ be probability measures on $\mathbb{R}$ and $\Pi(\mu_1, \mu_2, ..., \mu_d)$ be the set of probability measures on $\mathbb{...
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552
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Egorov's and Lusin's Theorem in the space with infinite measure
Both the fundamental Egorov's and Lusin's Theorem in measure theory are given on any measurable space $X$ whose measure is finite.
On the measurable space whose measure is infinite, does there ...
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Measure on infinite dimesional $L^p$ space relating size in norm to size in measure
Let $A$ be a bounded set in an infinite dimensional $L^p$ space. Fix an $\epsilon>0$. Is there a Borel measure $M$ such that
$$ M(B(x,\epsilon)) \geq C, \quad \forall x \in A$$
for some $C>0$ ...
- 111
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The Minkowski $(N-1)$- dimensional upper constant of a closed curve?
Let $\Omega\subset \mathbb R^N$ be open bounded smooth boundary. Let $S\subset \Omega$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<+\infty$. It is well know that if $S$ is not closed, then ...
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The density one properties of $\mathcal H^{N-1}$-rectifiable set
Let $S\subset \mathbb R^N$ be a $\mathcal H^{N-1}$-rectifiable set. Then we know that there exist countably many Lipschitz $N-1$-graphs $\Gamma_i\subset \mathbb R^N$ such that
$$
\mathcal H^{N-1}\...
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Can we represent a $N-1$ rectifiable set locally as a graph, with some price?
This is a follow up discussion for this post.
Let me copy part of the definition here.
Let $S\subset \mathbb R^N$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<\infty$.
My (updated) ...
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What does the following space look like?
Pick fixed $a=(a_1,a_2,\dots,a_d)\in\{\pm1\}^d$.
Consider map $F_a:\underbrace{\Bbb R^n\times\dots\times \Bbb R^n}_d\rightarrow\Bbb R^n$ given by $F(x_1,\dots,x_d)=\sum_{i=1}^da_ix_i$.
Denote $S_n\...
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Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?
The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question:
Given ...
- 1,771
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existence of locally translation-invariant Borel measure on Frechet manifolds
It is well known that the only locally finite, translation-invariant Borel measure on an
infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
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Measurable projection theorem
Hi ;
i have this theorem from the book :Set-valued analysis
Let $(\Omega,\mathcal{A},\mu)$ be a
complete $\sigma$-finite measure space
, $X$ a complete separable metric
space and $G\in\...
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