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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Does the centroid depend continuously on the curve?
Let $\gamma$ be a piecewise smooth curve in $\mathbb{R}^n$. Recall that the centroid of $\gamma$ is the point $(\overline{x}, \overline{y})$ where $\overline{x}$ is the average value of $x$ on $\...
- 29.2k
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votes
2
answers
3k
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Integration on the space of symmetric matrices
Let $\mu$ be a Lebesgue measure on the space $G$ of real symmetric $n \times n$ matrices (the Haar measure on the additive group of such matrices). For any $A \in G$ let $\chi_{A}(x)$ be its ...
- 1,329
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2
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3k
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Absolute continuity on $R^{n}$
I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$.
I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
- 399
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1
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978
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On a compact manifold, what kind of function can be the Jacobian of a diffeomorphism?
I could not answer or find references of this question, even for the following special case:
On $S^2$ (the two-sphere equiped with the standard Riemannian metric), is every positive smooth function ...
- 1,804
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3
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2k
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Origin of term Ahlfors-David regular
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...
- 565
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1
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Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
- 307
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1
answer
1k
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A strange Lipschitz function
Let $n \geq 3$. Does there exist a Lipschitz function $f: \mathbb R^n \to \mathbb R$ such that the following conditions hold?
The origin is a weak Lebesgue point of $\nabla f$, in the sense that the ...
- 6,155
10
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2
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496
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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 ...
- 119
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1
answer
893
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Doubling space without Besicovitch covering theorem?
A metric space is doubling if any ball of radius $2R$ can be covered by $N$ balls of radius $R$ and $N$ is fixed once forever.
Is there an example of complete length-metric space which is doubling, ...
- 293
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Current vs Varifold
I know the basic definitions concerning current and varifold, and they are generalization of submanifolds. What are their respective pros and cons? What are their crucial similarities and differences?
- 1,630
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Is there an inscribed cube for an arbitrary compact closed surface?
Given a compact closed surface $M$ (2-dim topological manifold) isometrically embedded in $\mathbb{R}^3$, are there 8 points $x_i\in M(i=1,\dots,8)$ such that they are the vertices of a cube $C\subset\...
user178596
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2
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301
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Continuity of length and area
Let $C_n$ be a sequence of rectifiable simple closed curves in $\mathbb{R}^2$ that converge to a rectifiable simple closed curve $D$ in the Hausdorff topology. It is easy to construct examples where
...
- 103
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How to shrink a square with minimal distortion?
$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\euc}{\mathfrak{e}}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\al}{\alpha}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\Lip}{\text{Lip}_{\text{inj}}...
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1
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Can a big set always look small?
For a set $C\subset \mathbb R^2$, define its visibility from a point $x$ as $vis_C(x)=\{\varphi\in \mathbb S^1\mid \exists t>0~~x+t*\varphi\in C\}$, where $\mathbb S^1$ denotes the unit circle.
Say ...
- 18.8k
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1
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Decomposability of Hausdorff measure
Consider $s$-dimensional Hausdorff measure $\mathcal{H}^s$ on the Borel sets in $\mathbb{R}^n$.
$\mathcal{H}^s$ is not $\sigma$-finite if $s < n$, but it is semifinite (on Borel sets!)
Is it known ...
- 2,049
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1
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463
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Isoperimetric inequality for closed curves in $\mathbb{R}^n$
A well known isoperimetric inequality for closed curves in $\mathbb{R}^2$ can be generalized to closed curves in $\mathbb{R}^{2n}$, see: https://mathoverflow.net/a/321505/121665.
I have two questions:
...
- 28k
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0
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802
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Topological dimension, Hausdorff dimension, and Lipschitz mappings
I can prove the following result. Here $\operatorname{dim} X$ stands for the topological dimension and $\mathcal{H}^n$ denotes the Hausdorff measure.
Theorem. Suppose that $f:\mathbb{R}^n\supset\...
- 28k
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464
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Reference for sets of locally finite perimeter on Riemannian manifolds
I am looking for a reasonably complete reference for Ennio De Giorgi's theory of sets of locally finite perimeter (also christened by him as Caccioppoli sets, after Renato Caccioppoli's pioneering ...
- 7,817
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Maximizing an integral w.r.t. a measure on the unit sphere
I would like to know if the answer to the following question is known.
Let $d \ge 3$. What is the value of
$$
\theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \...
- 980
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265
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Plank invariant measures on convex bodies
Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...
- 7,149
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Why do almost all points in the unit interval have Kolmogorov complexity 1?
Re-posted from math.stackexchange as I did not get any answers there.
I am reading
Jin-yi Cai, Juris Hartmanis, On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line, ...
- 297
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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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2
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877
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Concentration of measure for arbitrary convex bodies?
There are various "concentration-of-measure" theorems,
the best known that due to Lévy,
which is this (informally): the volume of a sphere $S^d$ in $d$ dimensions is largely
concentrated around an $\...
- 151k
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3
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934
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local behavior of a finite Borel measure
Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I recall,...
- 1,839
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1
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733
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Calderon-Zygmund decomposition on manifolds?
The classical Calderon-Zygmund decomposition says that if $f\geq 0$ is $L^1$ on a cubes $B$, with average value $\alpha$, then there is a sequence of disjoint cubes $B_j$, such that the average of $f$ ...
- 637
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349
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Tiling the plane with finitely many congruent pieces
Suppose $A_1,\dots,A_n$ are measurable subsets of the plane that are all related by rigid motions such that $|(A_1 \cup \dots \cup A_n)^c| = 0$ and $|A_i \cap A_j| = 0$ for all $1 \leq i < j \leq n$...
- 19.7k
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A Besicovitch-type Covering Theorem
In the book The Geometry of Domains in Spaces by Krantz and Parks, the authors proved the weak $(1,1)$-type estimate of the maximal function $M_\mu f$, where $\mu$ is a Radon measure, using their ...
- 1,245
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2
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695
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Non-calibrated area-minimising surface
Let $(M^{n+k},g)$ be a Riemannian manifold. Call a surface $\Sigma^n \subset M$ calibrated if there is a closed $n$-form $\omega$ defined on a neighbourhood $U \subset M$ of $\Sigma$ so that $\omega \...
- 5,038
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Measures whose projections are absolutely continuous
Since my question was not answered on MSE, I would like to ask it here.
Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt ...
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Is there a characterization of the Hausdorff measures?
It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
- 1,035
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2
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299
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Isoperimetric dimension for any (metric) measure space?
$\newcommand{\v}{\operatorname{vol}}$The isoperimetric dimension is the maximum $d$ s.t.
$$\v(D)\leq C\cdot \v(\partial D)^{d/d-1}$$
for all open with smooth boundary $D\subset M$, differentiable ...
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2
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common dominating measure for a family of measures
Given a family $\{\mu \}_{i\in I}$ on a Polish space (complete, separable metric space) $X$. When does there exist a measure $\lambda$ such that
$$\mu_i=f_i \lambda$$
where the $f_i$ are densities (...
- 1,256
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1
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492
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Dispersion points of Lipschitz functions
For a function $f: \mathbb R^n \to \mathbb R^m$ with $m < n$, we say that $x \in \mathbb R^n$ is a dispersion point of $f$ if
$$\liminf_{y \to x} \frac{|f(y) - f(x)|}{|y - x|} > 0.$$
Question: ...
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approximation of currents
Let $M$ be a closed Riemannian manifold of dimension $d$. Let $d \alpha$ be a smooth exact $p$-form. We define a current $T_{d \alpha}$ as follows : for any smooth $(d-p)$-form $\beta$ we set
$$ T_{d \...
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Existence of barycenter
Let $(X,d)$ be a metric space. A barycenter of a Borel probability measure $\mu$ on $X$ is a minimizer of the function
\begin{equation}
\begin{split}
f \colon X & \to \mathbb{R}\\
x &\mapsto \...
- 1,347
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3
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Almgren's mimeographed lectures notes on varifolds
I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to ...
- 1,616
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Metric measure spaces: in what sense is analysis on these spaces "non-smooth"
I understand the basic definition of a metric measure space to be the following:
A metric measure space is a triple of a space $X$, metric $d$, and measure $m$: $(X,d,m)$ in the sense that the ...
- 427
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How badly can the Lebesgue differentiation theorem fail?
Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is integrable. Is it true that
$$
\lim_{r\to 0}\frac{\displaystyle\int_{B_r(0)}f(y)~\mathrm dy}{r^{n-1}}=0 \quad ?
$$
This is obvious if $0$ is a Lebesgue point ...
- 1,149
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3
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287
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Set with small internal radius, small perimeter and prescribed area
Given a regular set $E\subset \mathbb R^2$ define
$$
R(E) = \sup\{r\colon \exists x,\ B(x,r)\subseteq E\}
$$
to be the radius of the largest circle contained in $E$ and let $|\partial E|$ be the ...
- 702
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1
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574
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On functions with strict Lipschitz constant
We say a measurable subset $S$ of $\mathbb R^n$ is measure dense if for every open set $U \subset \mathbb R^n$, $U \cap S$ is of positive Lebesgue measure.
Let $n \geq 2$, and let $f: \mathbb R^n \to \...
- 6,155
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1
answer
234
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Is there a non-atomic finite positive measure in the plane, of which uncountably many projections have atoms?
I would like to know whether or not there exists a finite probability measure $\mu$ on $\mathbb R^2$ which has no atoms, but such that there exists an uncountable set $A\subset \mathbb S^1$, such that ...
- 2,041
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What do singular, atomless invariant measures of $\times d$ look like?
Consider the circle map $\times d:x\mapsto dx \mod 1$. The lebesgue measure is the only absolutely continuous invariant probability measure, but this map has many other invariant measures. Of course, ...
- 14.4k
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1
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503
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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 \...
- 622
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1
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How large can the set of turbulent points be?
This question resisted attempts on MSE.
Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold:
$$\...
- 6,155
8
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1
answer
454
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Pseudo differentiable functions
Let $f: \mathbb R^n \to \mathbb R$ be a measurable function. Let $\mathcal L$ be the set of linear functions $\mathbb R \to \mathbb R$.
Define the roughness $\mathcal Rf(x)$ of $f$ at $x \in \mathbb R^...
- 6,155
8
votes
1
answer
865
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Fubini's theorem for Hausdorff measures
$B\subset \mathbb{R}^2$ is a Borel set. Define the slices $B_x:= \{y \in \mathbb{R}: (x,y) \in B \}$.
If $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$, presentations of Fubini's theorem often ...
- 675
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3
answers
804
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How to interpret this quote of Lin?
I recently stumbled across a quote of Fang-Hua Lin that I have trouble understanding [1, page 42].
It is a well-known fact that a weakly converging sequence of stationary integral currents may have a ...
- 5,038
8
votes
1
answer
513
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An isoperimetric-type inequality inside a cube
I am looking for a reference for the following inequality: if $\Omega \subset [0,1]^d$ satisfies $\mbox{vol}(\Omega) \leq 1/2$, then
$$ \mathcal{H}^{d-1}\left( \partial\Omega \cap (0,1)^d\right) \geq ...
8
votes
1
answer
441
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Axioms of length
Assume I want to define length of plane curves axiomatically.
It seems to be reasonable to assume that
The length of a unit segment is 1;
Congruent curves have equal lengths;
Length is additive with ...
- 45k
8
votes
2
answers
297
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Averaging the mass of a Sobolev function $f\in W^{1,p}(\Omega)$ near $\partial\Omega$
Recently, I asked a somewhat related question here. In the comment section, I found the formula
$$
\lim_{r\to 0}\frac{1}{r}\int_{\Omega_r} f(x)\,dx = \int_{\partial \Omega}f(\sigma)\,d\mathcal{H}^{n-1}...
- 1,245