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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Are Carnot groups ever CAT(𝜅) spaces?
Let $G$ be a free Carnot group of homogeneous dimension $d$, equipped with the Carnot–Carathéodory metric. Is $(G,d)$ ever $\operatorname{CAT}(\kappa)$ for some $\kappa\in \mathbb{R}$?
- 195
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98
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Lower bound estimate for the sum $\sum \text{diam}(U)^d$ over all countable covers of a cube
This question is inspired from the definition of Hausdorff measure. Let $C$ be a closed unit hypercube in $\mathbb R^d$ (side length equal to one, including boundary. The cube itself is at top ...
- 1,565
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199
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Absolute continuity of joint distribution if all marginals in any basis are absolutely continuous
Consider a probability distribution $\nu$ on $(x,y)\in\mathbb{R}^2$. I know that the absolute continuity of the marginals on $x$ and $y$ is not sufficient to imply the absolute continuity of $\nu$, ...
- 449
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Is it possible to define the trace of a function over a rectifiable set?
Let $\Omega$ be a bounded open set with smooth boundary and $E$ a set of finite perimeter in $\Omega$, i.e. $$P(E;\Omega)=\left\{\int_E\text{div}\: T\:dx:T\in C^\infty_c(\Omega;\mathbb{R}^n), |T|\leq1\...
- 121
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134
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Hausdorff dimension of a compact Lie group [closed]
Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant Riemannian metric and let $m$ be left-invariant Haar measure on $G$.
Now that $G$ is a metric space ...
- 323
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72
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Initial-boundary value problem for transport equation with $W^{1,p}$ velocity
Let us consider $v:\mathbb R_+ \times \mathbb R \to \mathbb R_+$ such that $v \in L^1(0,\infty, W^{1,p}(\mathbb R))$ and the transport equation
$$ \begin{cases}
u_t + v(t,x) u_x = 0 \qquad & (...
- 61
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141
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On the uniform boundedness principle and the space of functions of bounded variation
Let $U$ be a bounded smooth domain of $\mathbb{R}^d$. We write $m$ for the Lebesgue measure on $U$. A function $f \in L^1(U,m)$ has bounded variation in $U$ if
\begin{align*}
V(f,U):=\sup \left\{\int_{...
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Does a sequence of Jacobians converge to the 'correct' continuous part plus some controlled singular part?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \in W^{1,...
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Bound for the cardinality of maximal $r$-separable subsets contained in a ball of radius $R$ in $\mathbb R^d$
Let $B$ be a closed ball in $\mathbb R^d$ of radius $R$ and let $N=N_R(r)$ denote the maximal cardinality of the $r$-separated sets (meaning any two points in this set have distance at least $r$) that ...
- 1,565
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Restriction of Rademacher Complexity
Let $F\subseteq C([0,1]^n,\mathbb{R})$ be a finite family of functions, which is non-empty. Let $A,B$ be subseteq of $[0,1]^n$, again non-empty, and let $Rad(C)$ denote the Rademacher complexity of ...
- 5,405
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Perimeter decreases under intersection with half spaces
The main thing i need to prove is the following assertion:
Let $E\subset R^N$ be a set of finite perimeter and $H=\{x\in R^N : x\cdot e < t \}$ for $t\in R$ and $e\in S^{N-1}$.
Then prove that $$ ...
- 73
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Lebesgue dominated convergence for currents
I am learning current theory presently. In Chap 3.3-Definition of Monge-Ampère Operators, J.-P. Demailly, Complex analytic and differential geometry, I am a little confused as follows.
Let $X$ be a ...
- 391
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Measure in $\mathbb {C} ^p$ [closed]
If we have a non-constant holomorphic map $ f: \mathbb C ^ p \to X $, where $ X $ is a complex manifold. Let $ \omega $ be a metric on $X$, so $ \omega $ is a positive definite $ (1,1) $-form.
Is $ f ...
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Why $h_t$ maps into $\mathbb{R}^{\nu}$?
I am studying geometric measure theory (Herbert Federer - Geometric measure theory) and I have a question about class $r$ homotopies. Here's the definition, from p. 363, Section 4.1.9:
Suppose $U$ is ...
- 161
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Differentiation under the integral sign for a $L^1$-valued function (shape derivative)
Let
$d\in\mathbb N$;
$U\subseteq\mathbb R^d$ be open and $$\mathcal A:=\{\Omega\subseteq U:\Omega\text{ is bounded and open and }\partial\Omega\text{ is of class }C^{0,\:1}\};$$
$E:=\bigcup_{\Omega\...
- 167
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79
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Conditions for a function to vanish almost nowhere on its support?
Let $f:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function and $\mathrm{supp}(f) := \mathrm{cl}\{x\in\mathbb{R}^d\mid f(x)\neq 0\}$ its support.
Under which conditions is it true that $f≠0$ (...
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Are positivity for forms and that for currents consistent when talking about smooth forms?
Let $X$ be a complex manifold and $\theta$ a smooth $(1,1)$-form on $X$.
(1) If $\theta>0$ in the sense of currents, then can we deduce that $\theta>0$ in the sense of forms?
(2) If $\theta>0$...
- 391
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zero extension of positive currents are always positive
In Demailly's Complex Analytic and Differential Geometry page 139:
He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...
- 295
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60
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Compatibility of the Hausdorff measure with short exact sequences in normed spaces
Let $(E,\|.\|)$ be a finite dimensional normed space and take $F\subset E$ a
subpace, so that we have the canonical short exact sequence
$0\rightarrow F\rightarrow^\iota E\rightarrow^\pi E/F\...
- 313
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117
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Reference request: harmonic analysis with non-Lebesgue reference measure
The Lebesgue measure on $\mathbb{R}^d$
admits the following polar decomposition:
$$
L(dx) = r^{d-1} dr \lambda(dy),
$$
where 𝜆 is the uniform measure on the Euclidean unit sphere of $\mathbb{R}^d$ ...
user153086
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45
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Decomposition of the space of Radon measures with respect fractional harmonic capacity?
It is well know that there is a generalization of Lebesgue decomposition theorem in the following way:
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
- 371
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92
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Arithmetic product and sum of limit sets of non-elementary Fuchsian group of second kind
Let $L \subset \mathbb{R}$ be a limit set of a Fuchsian group $\Gamma$. If $\Gamma$ is a non-elementary Fuchsian group of second kind, then $L$ is a Cantor set. For example: $\Gamma= \bigg\langle \...
- 333
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183
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Total Mean Curvature as a integral on the whole space
It is well known from De Giorgi that we may express the surface area of a domain $\Omega\subset\mathbb{R}^N$ as:
$$
\int_{\partial\Omega} 1\ d\sigma=\int_{\Omega} ||\nabla H(\phi(x))||\ dx=\int_{\...
- 1,759
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100
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Weak estimate for difference quotient of BV function
In an answer to the question Weak Lebesgue spaces and an estimate for BV functions it was remarked that if $u\in BV(\mathbb R^N)$ then there exists a Lebesgue negligible set $F \subset \mathbb R^N$ ...
- 839
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If $u$ is $BV$ then $\operatorname{curl} Du = 0$ in the sense of distributions
Let $u\in BV(\mathbb{R}^N; \mathbb{R}^M)$. How does one prove that $$\operatorname{curl} Du = 0$$ holds in the sense of distributions?
- 839
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Consistency of the definition of total variation for functions of one or several variables
Where can I find a proof that the definition of total variation for functions of several variables is consistent with the definition of total variation for functions of one variable?
- 839
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107
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Level sets of a BV function and its derivative
Given $u \in BV(\Omega; \mathbb{R}^M)$, where $\Omega \subset \mathbb{R}^N$, what is the relationship between its level sets and its distributional derivative $Db$?
More specifically, does Alberti ...
- 839
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Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case
By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user123457
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105
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Gaussian isoperimetry for $\ell_p$ norms
Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$. It is well-known (e.g see Proposition 1) that for a given Gaussian volume content, half-spaces $H=\{x \in \mathbb R^n | a^Tx \le b\}$ ...
- 6,853
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How to show that this function is continuous (Geometric Measure Theory)
I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by
$$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$
is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...
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Is there a precise relationship between ``Geometric Functional Analysis" and high-dimensional probability/information theory?
The 2009 course on GFA by Roman Vershynin (https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf) introduced the subject with this line on the course page, "...
- 2,246
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Change of variables between quadrilaterals - Rayleigh quotient
A - Vertex at bottom left
B - Vertex at bottom right
K - Vertex at top left of blue quadrilateral
C - vertex at top left of brown quadrilateral
L - vertex at top right of blue quadrilateral
F - ...
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Comparing two $\sigma$-algebras
Let $X$ be a set. We denote $P(X)$ by the family of all subsets of $X$. We also denote $P(X)\otimes_{\sigma}P(X)$ by the $\sigma$-algebra generated by $\{A\times B: A,B \subseteq X\}$.
Q. For which ...
- 4,058
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Integral of the square of the areas of slices of a shape
Suppose $\omega$ is a bounded shape in $\Bbb{R}^3$ and that $\{z : (x,y,z) \in \omega \}=[0,T]$ (that is, the shape is exactly contained in the band $\{z \in [0,T]\}$. If we denote by $\omega_t = \{(x,...
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Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$
Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
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251
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Compact sets of Hausdorff dimension zero
I have a question about Hausdorff dimension. Suppose S is a compact subset of $\mathbb{R}^n$ whose Hausdorff dimension is zero. Does it follow that S can be covered by a finite DISJOINT union of ...
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Measure of the boundary of the support of a certain function defined by an expectation
Suppose:
$\mathcal{S} = \{ S \in \mathbb{R}^d \ | \ S_i > 0, \forall i = 1,...,d \} $
$R$ is a random vector (on some probability space, $\Omega$) such that, $R: \Omega \to \mathcal{S}$.
$h : ...
- 111
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326
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Approximation of Borel sets
Let $\nu$ be a finite Radon measure on $\mathbb{R}^2$ and denote the Lebesgue measure on $\mathbb{R}^2$ by $\mathcal{L}^2$. Assume that $\nu<<\mathcal{L}^2$.
We denote the boundary of $A\subset\...
- 347
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96
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Random projection increases the distance?
Consider two absolutely continuous random variables $X: \Omega \mapsto \mathbb{R}^d$ and $Y: \Omega \mapsto \mathbb{R}^d$ for probability spaces $(\Omega, \mathcal{F},p_X)$ and $(\Omega, \mathcal{F},...
- 482
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Ahlfors regular path metric defined by a continuous plane field in $\mathbb{R}^{3}$
Suppose I have a uniformly Holder continuous plane field $H$ on $\mathbb{R}^{3}$. I will assume that this plane field $H$ has many special properties, all of which are completely unreasonable to ...
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263
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Does a growing manifold fixed at a point converge to its tangent plane?
Let $M$ be a smooth compact $(n-1)$ dimensional submanifold in $\mathbb{R}^n$. Let $H$ be the $(n-1)$ dimensional Hausdorff measure. Let $f(x,y,t)$ is a function for $x\in\mathbb{R}^n$, $y\in\mathbb{R}...
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111
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Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?
I also put this question on MSE here
Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable).
Let $\...
- 679
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64
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decomposition of codim 1 currents
Let $T$ be an codimension one rectifiable mass-minimizing current. If $\partial T\llcorner B_R(a)=0$, then it is possible to write $T$ as an infinite sum of oriented boundaries: $T\llcorner B_R(a)=\...
- 43
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109
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Pointwise convergence of a sequence of approximate limits of BV functions
So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...
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112
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Volume of intersection of a convex polytope with general affine space
This question generalizes (this question) on the same site.
Let $\Delta^{n}$ denote the $n$-dimensional simplex in $n$ dimensions.
That is, $\Delta^{n}$ is the convex closure of the origin
and the $n$...
- 5,327
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157
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Laplace method with "bad" zero set
It is well-known that if $\phi:\mathbb{R}^n \longrightarrow \mathbb{R}$ is a function with $\phi(0) = 0$, $\phi(x)>0$ if $x \neq 0$ and $D^2\phi(0) \geq 0$, then the integral
$$\int_{\mathbb{R}^n} ...
- 9,853
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125
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$C^1$ Sard related question
Let $X$ be a $k+1$ rectifiable set with finite $k+1$ Hausdorff measure in $\mathbb{R}^{n+1}$ and set $Z=\{x\in X \mid e_{n+1}\perp T_xX \}$, where $T_xX$ is the approximate tangent and $e_{n+1}$ is ...
1
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1
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150
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Probability Content of a random ball in R^n
As a follow up to this question, concerning this paper:
Given random variables $X_1,\ldots,X_N,X_q:\Omega\rightarrow\mathbb{R}^d$, where $X_1,\ldots,X_N$ are independent and identical distributed. ...
- 329
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151
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Does Newtonian capacity increase strictly when mass is spread?
We start with two disjoint compact sets A and B with positive capacities. Then, we translate B s.t. $B+rv$ is disjoint from A and B and ,more importantly, $dist(x,y)<dist(x,y+rv)$ for all $x\in A$ ...
- 5,474
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83
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Convergent algorithm for dividing a body into two regions of equal volume
Let $\Omega \subset R^3$ be a bounded open region. It is well known that there exists a smooth surface $\Gamma$ with minimum area and constant mean curvature which is orthogonal to $\partial \Omega$ ...
