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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Is the Gaussian Correlation Inequality universal?
T. Royen proved the Gaussian correlation inequality in the context of Gamma distributions back in 2014, which was since popularized by Latala and Matlak. The properties of Gaussian integration seem ...
- 4,466
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254
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Characterizing surface area
(This question is a variant of an unanswered question at math.stackexchange.)
The Definition section of Wikipedia's article on surface area currently starts as follows:
While the areas of many ...
- 24.8k
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106
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Infering shapes from overlap with a shifting circle
A recent episode of Star Talk Radio discussed among other things the unknown object(s) orbiting Tabby's star (aka "Alien mega structure discovered!" in non-scientific media) and an astronomer said ...
- 9,650
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How do sets with unit fractional Hausdorff measure of dimension $>1$ look like?
Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask:
Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^...
- 12.7k
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332
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Monge-Ampère measures and Kazarnovskii pseudovolume
Let $\Gamma\subset\mathbb C^n$ be a convex polytope and let $h_\Gamma(z)=\max_{v\in\Gamma}{\rm Re}\langle z,v\rangle$ be its support function with respect to the standard scalar product on $\mathbb C^...
- 311
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For $f$ Lipschitz with $|\nabla f| = 1$ a.e., what is the supremal Hausdorff dimension of the set on which $\varepsilon< |\nabla f| < 1-\varepsilon$?
Let $f$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere.
Let $\varepsilon \geq 0$. What is the supremal Hausdorff dimension of the set on which $f$ is differentiable with $\varepsilon &...
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Invariance of the Lebesgue measure
It is well known that the Lebesgue measure is the unique (up to a multiplicative constant) sigma-finite Borel measure on $\mathbb{R}^d$ which is translation invariant.
I am wondering if a similar ...
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448
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Uncountable collections of distinct subsets of an interval (existence)
Throughout, $\mu$ is just the Lebesgue measure.
Question: does there exist an uncountable family of distinct subsets of $[-1, 1]$, denoted by $(U_j)_{j \in [-1, 1]}$, with $\mu(U_j) > 0$ for each $...
- 101
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3
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678
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How can dimension depend on the point?
Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
- 8,086
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The relation between Hausdorff dimension of an $n$-manifold and $n$
It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general.
For the case of manifolds, suppose $M$ is a $n$-manifold with a metric(distance)...
- 285
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547
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Maximal Hausdorff dimension of the set on which derivatives do not agree
Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the supremal Hausdorff dimension of the set on which $f$ and $g$ are both ...
- 6,155
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437
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Proper homotopy
Let $F: X \times [0, 1] \to Y$ be a homotopy such that for any $t \in [0,1]$ the map $F( \cdot, t) : X \to Y$ is proper. Is it true in general that $F$ is proper?
I am interested in particular in ...
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About taking an expectation over orthogonal matrices
Say $Q$ is a random variable which is sampling orthogonal matrices in $m$ dimensions using the Haar measure on $O(m)$. Let $A$ and $B$ be some (fixed) subset of rows and columns of $Q$ such that $\...
- 2,246
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Hausdorff dimension of the boundary of fibres of Lipschitz maps
Let $f: \mathbb{R}^m\rightarrow \mathbb{R}^{m-k}$ be a Lipschitz map.
Can we get a uniform estimate on the Hausdorff dimension of the boundaries of fibres of $f$? I.e. do we have an upper bound for
...
- 1,045
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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 $j$;
...
- 1,179
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Geometric Construct for Integrating Symmetric Tensors?
I'm interested in finding the appropriate geometric construct for the integration of symmetric tensors, analogous to the way differential forms can be integrated over manifolds.
The motivation comes ...
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Measure of chords from a cantor set
The following problem is inspired by a problem in Pugh's Mathematical Analysis book. (Chapter 2 Problem 42).
In the problem he asks one to consider the standard Cantor set on the unit interval, and ...
- 1,187
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Higher (BV) regularity of solutions to Poisson equation with Radon measure right-hand side?
I am trying to understand higher regularity for solutions to Poisson's equation when the right-hand side is a Radon measure. In particular:
$$\begin{cases}
\Delta u = \mu \text{ in } \Omega\\
u = 0\...
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Exotic homeomorphisms of a cube
If $\varphi:\mathbb{R}\to\mathbb{R}$ is continuous, non-constant, non-decreasing, and differentiable a.e. with $\varphi'=0$ a.e., then the mapping
$$
\Phi(x,y)=(x+\varphi(x),y+\varphi(y))
$$
is a ...
- 28k
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Isoperimetric type inequality in $\mathbb{R}^2$
Fix $L \in (0,\infty)$ and consider $\mathcal{C}_L$ defined as follows:
\begin{align*}
\mathcal{C}_L := \{ \gamma:[0,1] \rightarrow \mathbb{R}^2 |~ \gamma \text{ is smooth and length($\gamma$)$=L$ }\}....
- 399
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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 ...
- 459
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504
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Anisotropic perimeter and regularity of anisotropic minimal surfaces
1. Introduction.
By-now classical results assert that minimal surfaces (in $\mathbb R^n$) are generically "smooth" out of a "small" set.
Question. What are the known regularity results for ...
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Generalization of area and coarea formula for fractional Hausdorff measures
Let $X,Y$ be Polish spaces, $s,t>0$ and $F:X\to Y$ locally Lipschitz continuous such that $X$ is $\sigma$-finite w.r.t. the $(s+t)$-dimensional Hausdorff measure $\mathcal{H}^{s+t}$.
The Eilenberg ...
- 9,650
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Crofton formula: expected intersections is to length as variance is to what?
There is this beautiful Crofton formula for the length $L(C)$ of
a curve $C$ on the round unit 2-sphere: you take the expected number
of intersections of $C$ with a random great circle and multiply
by ...
- 7,005
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299
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Convexity of Isoperimetric Domains
I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive ...
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434
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Vector measures as metric currents
Currents in metric spaces were introduced by Ambrosio and Kirchheim in 2000 as a generalization of currents in euclidean spaces. Very roughly, a principle idea is to replace smooth test functions (and ...
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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}$$
then, for every ...
- 51
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Transportation-cost inequality for pushforward measure
Let $X=(X,d_X)$ and $Y=(X,d_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies ...
- 6,853
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953
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Precise density estimates for Cantor sets
Let $C_\lambda$ be the classical Cantor set associated to a real number $0<\lambda<\frac{1}{2}$, as defined for example in the book of K. J. Falconer The geometry of fractal sets. I recall ...
- 1,616
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Algebraic geometric measure theory
Suppose I have $V\subset \mathbb{C}^n$ be the zero set of a polynomial $P(z_1, \dotsc, z_n),$ with bounded height of coefficients (where height is, to fix something, $|\log|a||$) and degree $d.$ ...
- 96.4k
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A selection principle in measure theory
A Borel subset $B$ of the unit interval $\mathbb I=(0,1)$ is defined to be a density neighborhood of a set $A\subseteq\mathbb I$ if for every $a\in A$ we have $$\lim_{\varepsilon\to0}\frac{\lambda(B\...
- 41.8k
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An isoperimetric type of inequality in terms of Wasserstein distance/Optimal transport
Let $A \subset \mathbb{R}^n$ be a region having the same volume as an $n$ dimensional ball $B^n_R$ with radius $R$ centring at the origin.
Isoperimetric inequality says:
$ Vol_{n-1} \partial A \geq ...
- 365
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How to estimate the pressure?
I have a finite collection of diffeomorphisms $g_1,\cdots,g_n$ taking the unit interval $I$ to disjoint subintervals $I_1, I_2,\cdots,I_n$. If $G$ is the semigroup they generate, the limit set of $G$ (...
- 2,562
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A locally compact, complete metric space in which the closure of open balls coincide with the closed ball is Heine-Borel
I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces:
Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R &...
- 99
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Applications of the co-area formula
Kirchheim [2] generalized the classical area formula to the case of Lipschitz mappings into metric spaces. Ths paper is well known and widely cited. The area formula is a special case of the co-area ...
- 28k
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When are Lipschitz functions dense in continuous functions?
Let $X$ be a compact metric space, and let $Y$ be another metric space.
I am looking for examples of, and especially references to, theorems that give conditions under which any continuous mapping $f:...
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Level sets of weakly differentiable funtions
Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose
$$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$
where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...
7
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Hausdorff dimension and sigma finiteness
If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$.
I would like to see an example of such a ...
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656
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Minimal surface which divides a convex body into two regions of equal volume
Question. Given a convex body $\Omega$, what is the shape of a surface $\Gamma$ of minimal area which divides $\Omega$ into two regions of equal volume?
Background/motivation.
A 2D version of the ...
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Fractals of dimension zero
Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0?
I can think of something ad hoc like a Cantor middle $\frac13$ set where the ...
- 24.8k
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3
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700
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How to estimate the integral involving the distance function
Let $\Omega\subset\mathbb{R}^n$ be an open bounded domain with smooth boundary. Consider the following integral:
$$I(t)=\int_{\Omega}e^{-\frac{d^2(y,\partial\Omega)}{t}}{\rm d}y.$$
My problem is how ...
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Hausdorff dimension of convex set in ${\bf R}^n$
I want to know the smoothness of convex set in ${\bf R}^n$.
Recall the following definition.
Definition : $X$ is a bounded closed convex set in ${\bf R}^n$ if for $x$, $y\in X$, the any $d$-...
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Stability of minimal surfaces
Let $\Gamma$ be a prescribed $n-2$ dimensional set and assume $S \subset R^n$ is a minimal hyper-surface with respect to some smooth metric $g$ on $R^n$, and $\partial S= \Gamma$. Is $S$ is stable ...
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483
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Can Hausdorff dimension make sets into a Tropical Semiring?
If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure
\begin{equation}
H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : \...
- 1,104
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1
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Relationship between Erdos and Falconer distance problems
Given a set $E \subset \mathbb{R}^d$, define the distance set of $E$
$$
\Delta(E) = \lbrace|x-y| : x,y \in E \rbrace,
$$
where $|\cdot |$ is the usual Euclidean distance.
$\bullet$ The Erdos-...
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532
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If the measure theoretic boundary is closed must it coincide with the topological boundary?
$\DeclareMathOperator\Int{Int}\DeclareMathOperator\Ext{Ext}$Suppose $E\subset\mathbb{R}^n$ is a set of finite perimeter and suppose that the measure theoretic boundary $\partial^*E=\mathbb{R}^n\...
- 1,149
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Background for Varifold theory
I noticed this question posted on MO, hence I estimated that this may be an acceptable question even in MO (and not for MSE). I studied the notion of current and in a nutshell I understood "...
- 327
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A better version of Weyl's Law or uniform estimates of Laplacian higher eigenvalues
Let $(M^n,g)$ be a closed $n$ dimensional Riemannian manifold with $\mathrm{Ric}_g\ge -K$, $(K\ge 0)$. Weyl's law(along with Karamata Tauberian Theorem) asserts that the eigenvalue $\lambda_i$ of $-\...
- 193
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2k
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Derivative of distance function to a closed, rectifiable set
Let $\Gamma \subset \mathbf{R}^d$ be a closed, countably $n$-rectifiable set. Is there any reasonable way to write the derivatives
$$
\frac{\partial}{\partial x_i} \mathrm{dist}\, (x,\Gamma)
$$
for $x ...
- 1,179
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votes
3
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541
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About the Hausdorff dimension of removable singularities of PDE
There are some interesting phenomenons about removable singularities (or extension problems).
In the theory of functions of several complex variables, we know the classical Hartogs theorem:
Let $f$ ...
- 181