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.
717
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Converse on the rectifiability of products of rectifiable sets
Let $1\leq k\leq m$ and $1\leq l\leq n$ fixed integers, $\mathscr{H}^k$ the $k$ dimensional Hausdorff measure and $E\subset \mathbb{R}^m$. We say that :
(1) $E$ is $k$ rectifiable if there exists $C\...
- 1,588
1
vote
0
answers
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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0
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338
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When is Hausdorff measure locally finite?
Let $(X,d)$ be a metric space and let $H^\alpha$ denote the $\alpha$-dimensional Hausdorff Borel measure on $X$, where $\alpha$ is the Hausdorff dimension of $X$. Are there any simple conditions on $X$...
- 270
5
votes
1
answer
700
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Hausdorff approximating measures and Borel sets
Suppose $ 1 \leq m \leq n $ are integers and for each $ 0 < \delta < \infty $ let $\mathscr{H}^{m}_{\delta} $ be the size $ \delta $ approximating measure of the $ m $ dimensional Hausdorff ...
- 541
14
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0
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482
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Lebesgue density 1/2 (or bounded away from 0 and 1)
From the work of Preiss, we know that in infinite-dimensional spaces, one has violations of the Lebesgue density theorem. In particular, he has constructed examples of probability spaces where a set ...
- 6,061
2
votes
1
answer
309
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Is the $L^p$ space of tensors complete?
On a Riemannian manifold $(M,g)$, let $\mathcal L^p(M,k)$ denote the space of measurable $k$-tensors $T$ (i.e., the coordinate components in any chart are Lebesgue measurable) for which the norm
$$||...
- 728
4
votes
1
answer
352
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Locally doubling measures
Let us say that a measure $\mu$ on $\mathbb{R}^d$ is locally doubling if for each
$x\in\mathbb{R}^d$ there is a constant $C(x)$ such that for all $r>0$,
$\mu(B(x,2r)) \le C(x) \mu(B(x,r))$,
where $...
- 6,061
8
votes
1
answer
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 ...
- 9,539
4
votes
1
answer
721
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Lebesgue-Besicovitch theorem for partition elements rather than balls
I'll state the classic result in its density (rather than the more general differentiation) version. Let $\mu$ be a measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ and $A\subset \mathbb{R}^n$ ...
- 6,061
7
votes
1
answer
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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 ...
- 9,539
4
votes
1
answer
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Is a minimal surface $S$ that is bounded by an analytic closed curve $C$, analytic?
Let $C$ be an analytic closed curve (in the form of an unknot) in $\mathbb{R}^3$ and let $S$ be a minimal surface (a disk) bound by $C$. Is $S$ always analytic? Can you point out some references?
- 415
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2
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328
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a modification on an infinite Bernoulli convolution
The distribution $\nu_{\lambda}$ of the random series $\sum\pm\lambda^n$ is the infinite convolution product of $\frac12(\delta_{-\lambda^n}+\delta_{\lambda^n})$. This problem has been studied ...
- 41.8k
2
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1
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223
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Are there many "cusps" in a rectfiable star-shaped set?
Let me first recall the definition of density with respect to a measurable set $E$ as follows:
A point $x \in \mathbb{R}^n$ is a point of density $\alpha$ for $E$ if
$$\lim_{r \rightarrow 0} \frac{...
- 1,320
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votes
0
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76
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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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0
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45
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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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2
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Random packing density of cylinders in a volume
I am trying to calculate the packing density of cylindrical bottles in a box, assuming that the bottles are randomly dumped in the box.
I have read on the packing density of spheres here https://en....
- 133
3
votes
1
answer
137
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Bilipschitzian maps and densities
Let $ A \subseteq \mathbf{R}^{m} $ and suppose that $ \mathbf{R}^{m} \setminus A $ has $ m $ dimensional density equals $ 0 $ at a point $ a \in A $. Let $ B \subseteq \mathbf{R}^{m} $ and let $ f : A ...
- 541
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0
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89
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Asymptotical control of the measure of tubes covering subsets of fixed Hausdorff dimension
(A version of this question was posted on math stack exchange)
Let $M$ be a $C^1$ submanifold of dimension $n$ of $\mathbb{R}^N$, and denote $\mu$ the standard surface measure on $M$.
Consider a ...
- 975
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votes
2
answers
234
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Measures with finite mass relative to a fixed measure
Fix a function $f\in L^1_\text{loc}(\mathbb{R}^n)$. Let
$$
L^1_\text{rel}[f]=\{ g\in L^1_\text{loc}(\mathbb{R}^n) : \|g-f\|_1<\infty\}.$$
be space of functions which differ from $f$ by an $L^1$ ...
- 2,279
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votes
1
answer
122
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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 ...
- 701
3
votes
1
answer
870
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Stokes theorem for manifolds with boundary as disjoint union of submanifolds
Looking at the generalizations of Stokes theorem, I did find a version for manifold with corners, but I was surprised this generalization doesn't contain a simple example such as the cone. So my ...
- 549
5
votes
1
answer
411
views
Continuous deformation of soap films
Let $S$ be a soap film bounded by an unknotted wireframe cycle (in $R^3$). Why is it the case that as we deform the wireframe in $R^3$, $S$ deforms continuously?
- 51
2
votes
1
answer
372
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On the surface area of a set
I have a question about an estimate of the surface area of a set.
Let $B(r)$ denotes the open ball of $\mathbb{R}^{d}$ centered at origin with radius $r>0$. Let $F:\mathbb{R}^{d} \to \mathbb{R}^{d}...
- 701
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votes
5
answers
1k
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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 ...
- 24.6k
4
votes
1
answer
394
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Is the lower Minkowski content additive in any sense?
Proposition 3.3.2 in the book The geometry of domains in space by S. Krantz and H. Parks states that if the sets $A$ and $B$ are separated by a positive distance, then $\mathcal{M}_*^K(A \cup B) = \...
- 1,431
3
votes
2
answers
215
views
More refined versions of Brunn–Minkowski inequality and/or Prékopa–Leindler inequality
Brunn-Minkowski inequality lower bounds the measure of a Minkowski sum by the measures of the summands. Its statement reads as follows:
Let $n$ ≥ 1 and let $μ$ denote the Lebesgue measure on $\...
- 449
2
votes
0
answers
99
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Relationship between weight of spanning tree in a tree metric approximation and the original metric
So suppose we have a tree metric which approximates the Euclidean distance between a finite set of points. The leaves correspond to points in the original space. It may be an ultra metric, and ...
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12
votes
1
answer
516
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Can $C^1$ mappings with derivative of low rank be approximated by smooth maps?
Asked once on SE-mathematics.
Let $U$ be an open subset in $\mathbb{R}^n$, $m\in\mathbb{N}$, $1\leq m<n$ and let
$$\mathcal{C}^k_{\leq m}(U,\mathbb{R}^n):=\lbrace g\in\mathcal{C}^k(U,\mathbb{R}^n)\...
- 123
8
votes
1
answer
320
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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
4
votes
1
answer
945
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continuous injective extension of a map defined on a hemisphere
Let $S^2 = \{x\in \mathbf R^3\colon |x|=1\}$ be the unit sphere, $S^2_+ = S^2 \cap \{x_3 \ge 0\}$ be the upper hemisphere and $S^1 = S^2 \cap \{x_3 = 0\}$ be the unit circle. Let $u\colon S^2_+\to \...
- 692
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votes
1
answer
1k
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Doubling metrics, doubling measures, Lebesgue density
As stated in this question,
Lebesgue differentiation theorem holds on locally doubling space?
and proved here,
http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf
the Lebesgue differentiation theorem (...
- 6,061
7
votes
1
answer
438
views
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 ...
- 73
3
votes
2
answers
348
views
Polar coordinates, bounded domain with $C^{1}$ boundary
I have a question about a integral on a surface.
It is well known that for any Integrable function $f$ defined on $\mathbb{R}^{n}$, it holds that
\begin{equation}
(1) \quad \frac{d}{dr} \int_{B(0,r)}...
- 701
1
vote
0
answers
110
views
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
2
votes
1
answer
612
views
Results for Hausdorff Measure after Linear Transformation
For the Sierpinski Triangle, $S$, the $d$ dimensional Hausdorff measure is given by, $H^{d}(S)$. If a linear transformation, $W$ is applied to $S$, with
$$W(x,y)=\begin{bmatrix} 1/2 & 0 \\ 0 &...
2
votes
1
answer
160
views
harmonic differential form integer class
Let $(M,g)$ be a compact Riemannian three-fold such that $H_2(M,\mathbb{Z}) = \mathbb{Z}$ and $S$ any surface representing 1. By Hodge theory there exist a harmonic differential one-form $\eta$ dual ...
user50806
5
votes
1
answer
232
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sequence of graphs converge in the sense of varifold to multiplicity 2 plane
Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) ...
- 49
2
votes
0
answers
78
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Compute Mixed Volume with Respect to Some Regular Sets
Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
- 1,127
3
votes
1
answer
179
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Equivalent Definitions of the Gaussian Surface Measure for Regular Sets
I wonder if the following definitions of the Gaussian surface measure are equivalent.
First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....
- 1,127
4
votes
0
answers
386
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Topology on the space of Borel measures
Let $ B $ be the set of all measures $ \phi $ of $ \mathbf{R}^{n} $ such that every open set is $ \phi $-measurable (sometimes these measures are called Borel measures). Note the measures in $ B $ are ...
- 541
0
votes
0
answers
75
views
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
$$
\mathcal H^{N-1}\...
- 679
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votes
1
answer
338
views
Level sets of weakly differentiable funtions
Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose
$$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$
where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...
3
votes
0
answers
172
views
question about currents
I have a question in the field of currents:
Let M be an n-dimensional smooth manifold, and let T be a k-current (induced by a k dimensional sub-manifold), I would like to approximate it by a series of ...
- 31
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votes
0
answers
113
views
Does the Hodge *-operator act on the tangent space at 0 to the space of integral (n-1)-cycles in a conformal manifold of dimension d=2n?
Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$.
Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral
$k$-currents in $M$
and write ${\cal D}^{\mathit{int}}...
- 351
2
votes
0
answers
113
views
Currents with mean curvature
so let $T=\tau(M,\theta,\xi)$ a rectifiable current in $U\subset\mathbb{R}^{n+k}$, then $T$ has mean curvature vector $H$ if for every $X\in C^1_c(U\setminus\partial T,\mathbb{R}^{n+k})$ the variation ...
- 43
14
votes
0
answers
627
views
Are harmonic mappings non-singular outside a set of measure zero?
Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb D^n$.
Let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving immersion, and let $\omega :\...
- 6,621
1
vote
1
answer
247
views
Existence of stationary tangent cones
My question refers to Leon Simon's Book: Geometric Measure Theory, Chapter 42
So let $V\in G_n(U)$ a general Varifold and $\|\delta V\|$ the Total Variation measure. If the density $\theta^n(\mu_V,x)&...
- 43
1
vote
0
answers
63
views
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
8
votes
1
answer
250
views
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 ...
- 24.6k
1
vote
0
answers
107
views
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 $BV(\Omega)$. ...
