All Questions
Tagged with mg.metric-geometry measure-theory
69 questions
1
vote
1
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183
views
Metric currents on singular measures in $\mathbb R^d$
Unless I am misunderstanding a lot of works, it is my understanding that a finite and non negative measure $\mu=g\mathcal{H}^\alpha$, where $\mathcal{H}^\alpha$ is the $\alpha$-Haudorff measure, ...
- 391
2
votes
1
answer
246
views
Ramsey type property of the Lipschitz constant
The following problem was proposed by Pietro Majer as an extension of an earlier question of mine on Lipschitz functions.
For $f$ a Lipschitz function on $\mathbb R^n$, we denote by
$$\text{Lip}(f, U) ...
- 6,165
4
votes
1
answer
275
views
Lower bound on volume of $n$-cube intersected with $n$-sphere
Let $B_n^r(c)$ be the radius $r$ ball in $\mathbb{R}^n$ dimensions centered at $c$. I am interested in
$$\text{Vol}([-0.5, 0.5]^n \cap B_n^r(c)).$$
Is there a good lower bound for this quantity?
I was ...
- 75
3
votes
1
answer
199
views
Product of low dimensional Hausdorff measures
Let $\mathcal{H}^n$ and $\mathcal{H}^m$ be Hausdorff measures on $\mathbb{R}^n$ and $\mathbb{R}^m$. We know that the product measure $\mathcal{H}^n\otimes \mathcal{H}^m$ is the Hausdorff measure $\...
- 73
3
votes
1
answer
156
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Concentration of measure on spheres with respect to a unitary of trace approximately zero
Cross-posted from MSE, where it hasn’t received any answer yet:
This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
- 2,830
11
votes
0
answers
488
views
Are there 100 points that are part of every half-density part of the plane?
Is there a configuration $P$ that consists of 100 points of the plane such that every $X\subset\mathbb R^2$ whose density is half contains an isometric copy of $P$?
I am deliberately being vague ...
- 18.9k
2
votes
0
answers
151
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$\mathscr{H}^{n-2}(\Sigma)< \infty$ implies $\mathscr{H}^{n-1}(\pi(\Sigma))=0$
Let $\Sigma\subset \mathbb{R}^{n+1}$ be a set with $(n-2)$-dimensional Hausdorff measure finite, i.e. $\mathscr{H}^{n-2}(\Sigma)<\infty$. Let $\pi:\mathbb{R}^{n+1}\to \mathbb{R}^n$ be the ...
- 1,149
1
vote
1
answer
133
views
In the limit, do the intersection points of a string figure define a probability measure on the unit disk?
Let D = {z ∈ ℂ | |z| ≤ 1} denote the closed unit disk in the complex plane.
For any integer n ≥ 1 define the nth string figure S(n) ⊂ D as the union of all n(n+1)/2 line segments that connect two ...
- 2,858
25
votes
1
answer
3k
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A gerrymandering problem - can you always turn a tie into a landslide victory?
Note: Here we use $|A|$ to denote the Lebesgue measure of a measurable subset $A$ of $\mathbb R^2$.
Your party is running for election! In your country, voters are approximately uniformly distributed. ...
- 6,165
2
votes
1
answer
300
views
If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected
Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
- 1,149
4
votes
0
answers
169
views
Finding balls with big measure
Let $(X,d)$ be a compact metric space $n \in \mathbb{N}$ and $\mu$ a finite Borel measure. Suppose there exists $\delta, R>0$ such that for all $0<r<R$.
$$\mu(B(x,r)) < \delta r^n.$$
Under ...
- 41
7
votes
2
answers
180
views
Bisector of two points in a Riemannian manifold has measure $0$
Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$?
I was ...
- 10.6k
3
votes
0
answers
148
views
Can the Banach-Tarski paradox or Tarski's circle-squaring problem be done with hinges?
It is known for both the Banach-Tarski paradox and Tarski's circle-squaring problem that you can finitely partition the starting configuration, then continuously move these pieces (without ...
- 700
1
vote
1
answer
318
views
What is the limit of a helix as the frequency tends to infinity?
Consider the helix parametrized by $r(t) = (\cos(\omega t), \sin(\omega t), t)$, for a given $\omega > 0$, and $t \in \mathbb{R}$. How can we interpret the limit as $\omega \to \infty?$
My initial ...
- 217
6
votes
3
answers
532
views
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
5
votes
1
answer
430
views
Volume of a shape whose boundary consists of portions of spheres symmetrically placed about the origin in $d\gg 1$ dimensions
We are given a convex shape $S$ in the $d$-dimensional Euclidean space, whose boundary is formed by portions of $2d$ different spheres, one portion per sphere. The radius of each sphere is the same, $...
- 1,677
25
votes
3
answers
945
views
Are there arbitrarily large families of lines in $\Bbb R^3$ with average angle $\ge \pi/3$?
Question: Can I have an arbitrarily large finite family of lines $\ell_1,\dotsc,\ell_n\subset\Bbb R^3$ so that the average angle between two (distinct) lines is $\ge \pi/3$?
We can assume that all ...
- 13.6k
1
vote
1
answer
174
views
Dimension-preserving non-linear map
Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous non-linear map, and let $A$ be a connected subset of $\mathbb{R}^n$ with $\text{dim}(A)=d\leq n$. When can we say that the dimension of the image, $\...
- 39
5
votes
1
answer
266
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Contracting a set to a ball
$\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$
Question 1: Let $S$ be a nonempty measurable subset of $\R^n$. Let $B$ be a closed ball in $\R^n$ such that $m(B)=m(S)$, where $m$ is the Lebesgue ...
- 128k
6
votes
1
answer
551
views
Relationship between doubling constant of a metric space and of a metric measure space
Let $(X,d,m)$ be a metric measure space. We say that it is doubling in the sense of metric spaces if for every:
$x\in X$ and every $r>0$ there exists some (metric) doubling constant $C_d\geq 0$ ...
- 195
0
votes
1
answer
189
views
Terminology "upper" Ahlfors regular measure
Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for ...
- 5,405
5
votes
2
answers
245
views
Differentiability of the map $x\mapsto \delta_x$ in the Arens-Eells/Lipschitz-free space
$\DeclareMathOperator\AE{AE}\DeclareMathOperator\Lip{Lip}$Let $\AE(X)$ denote the Arens-Eells space on a Banach space $X$. Consider the map:
$$
\begin{aligned}
\delta: X & \rightarrow \AE(X)
\\
x&...
3
votes
0
answers
137
views
Isoperimetric inequality for general metric space
Consider some space $\mathcal{S}$ with metric $d$ and measure $\mu$.
For arbitrary set $H$ denote the $v$-bound of $H$ by $\delta_v(H):= \{x \mid x \notin H: \exists y \in H \text{ s.t. } d(x,y) \le v ...
- 2,576
5
votes
3
answers
359
views
Recovering the length metric from Hausdorff measure
The metric cannot be recovered from its Hausdorff measure in general. Now, assume that $(X,d_X)$ and $(Y, d_Y)$ are connected compact length spaces and induce $n$-dimensional Hausdorff measures $\...
- 1,799
1
vote
1
answer
138
views
Least square assignment and hyperplanes
Let $S$ be a finite set of points in $\mathbb{R}^{d}$, $c(s) \in [0,1]$ such that $\sum_{s \in S} c(s) = 1$, $\rho$ continuous and non-vanishing probability distribution on $[0,1]^{d}$ and $\mu $ ...
user178738
2
votes
0
answers
186
views
Metric on space of Borel-measurable functions
Let $(X,d_X),(Y,d_Y)$ be metric spaces and $X$ is locally-compact and fix a Borel probability measure $\nu$ on $X$. For any Borel-measurable $f:X\rightarrow Y$, let $\mathcal{K}(f,\delta)$ be the set ...
3
votes
1
answer
155
views
Distance function and geometry of the set
Let $X \subseteq \mathbb{R}^n$ be a closed $d$-dimensional regular set (i.e. for any $x \in X$ and $0<r< \text{diam(X)}>$, $\mathscr{H}^d(B(x,r)) \sim r^d$ ) which has the property that for ...
- 65
1
vote
1
answer
1k
views
Closed-form upper-bounds for Wasserstein distance between finite measures
Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \...
- 5,405
1
vote
0
answers
94
views
Recursive expression of Lebesgue measure for balls with removed intersection
This is not the most theoretically challenging question; rather it is more of a reference request for a simple formula (which must be known).
Let $\left\{B_{\epsilon_n}(x_n)\right\}_{n=0}^N$ be a set ...
- 5,405
-1
votes
1
answer
112
views
Isometric stratification preserves volume?
Let $K\subset \mathbb{R}^k$ be a non-empty compact subset let $f:K \to K$ be Lipschitz and surjective. If, moreover, $f$ is an isometry then clearly $f$ preserves the Lebesgue measure of $K$.
I ...
- 5,405
3
votes
0
answers
222
views
Sets of finite perimeter: intersection with an half space
I have a question regarding sets of finite perimeter. In particular I'm interested to find
$$\mu_{E \cap H_t}, \label{1}\tag{1}$$
where $E$ is a set of finite perimeter in a generic open set $\Omega \...
- 51
1
vote
0
answers
184
views
Bounding the total variation metric between Gaussian mixtures
Let $\mathcal{P}(\mathbb{R}^d)$ the space of probability measures on $\mathbb{R}^d$ with total variation metric $\delta$, fix $k \in \mathbb{N}$, and let $\mathcal{P}'\subset \mathcal{P}(\mathbb{R}^d)$...
- 5,405
10
votes
2
answers
496
views
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
4
votes
2
answers
525
views
Packing a Riemannian manifold with disjoints balls
Let $M$ be a smooth Riemannian manifold with Riemannian measure $\mu$. I don't suppose that $M$ is complete. Can we find a finite or countable disjoint collection of open (or closed) and relatively ...
- 121
11
votes
3
answers
565
views
Is Stoch enriched in Met?
Let $\mathsf{Stoch}$ denote the Kleisli category of the Giry monad. That is, $\mathsf{Stoch}$ is a category whose objects are measurable spaces and for which a morphism $f\in\mathsf{Stoch}(X,Y)$ is a ...
- 8,659
2
votes
2
answers
177
views
Measure of random Voronoi cell
Let $\mu$ be some distribution (with density) on $\mathbb{R}^d$, from which we independently draw $X_1,\ldots,X_n$. These induce a Voronoi partition on $\mathbb{R}^d$: $V_1$ is the set of all points ...
- 6,541
3
votes
1
answer
326
views
How can the same polytope have three different volumes? [closed]
I'm quite new to geometry and I came across the idea that the same convex polytope can have at least three different volumes.
Consider the permutohedron, formed by the convex hull of the n! points ...
- 163
2
votes
1
answer
261
views
Bounded ball measure on compact metric space
Fix $c>1$. Let $(X,d)$ be a separable compact metric space, does there necessarily exist a Borel probability measure $\nu$ on $(X,d)$ such that
$\operatorname{sup}_{x \in X,r>0}\frac{\nu(\...
- 5,405
10
votes
1
answer
643
views
Estimation of the Gromov–Wasserstein distance of spheres
Let $(X,d_X,\mu_X)$ and $(Y,d_Y,\mu_Y)$ be two metric measure spaces. A probability measure $\mu$ over $X\times Y$ is called a coupling if $(\pi_1)_\sharp \mu=\mu_X$ and $(\pi_2)_\sharp \mu=\mu_Y$. We ...
- 1,565
12
votes
2
answers
2k
views
How to think about dual space of a certain space of Lipschitz functions
Consider the following Banach space (for concreteness):
$$X=Lip(\bar{\mathbb{B}}^n)=\{f\in C^0(\bar{\mathbb{B}}^n): \Vert f \Vert_L<\infty \}$$
where
$$
\bar{\mathbb{B}}^n=\{\mathbf{x}\in \mathbb{...
- 2,478
2
votes
0
answers
144
views
Lebesgue density theorem for "doubling uniformly covering collections of subsets"
I am looking for a version of Lebesgue density theorem that works when restricting to "good" collections of balls with respect to (not necessarily doubling) metric measure spaces. Specifically
Let $(...
- 2,964
7
votes
2
answers
665
views
Non-separable metric probability space
Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if:
the support of $\mu$ is contained in a separable subspace of $X$.
Questions:
1. Is there a standard name for this property?
...
- 6,541
3
votes
0
answers
47
views
Measure of set of vectors whose outer product are bounded
Let consider the canonical Euclidean space $E = \mathbb{R}^n$, endowed with the Lebesgue measure $\mu$.
Define the map $v_k: E^k \rightarrow \mathbb{R}$ that
sends a $k$-uple $x_1,\cdots, x_k$ of ...
- 313
4
votes
0
answers
756
views
Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures
Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...
- 1,675
1
vote
1
answer
135
views
Is volume of abstract polytope realisation bounded by length of edges?
Suppose we have abstract polytope $F$ of dimension $d$ (that is the greatest rank facet has rank $n$). Such abstract object may have realisations in d-dimensional Euclidean space as polytopes $A_i(F)$,...
- 1,626
3
votes
0
answers
126
views
Does every non-locally compact metric space admit a violation of Lebesgue's theorem?
From the results of Preiss and Tišer, it is known that many natural families of measures on Hilbert spaces violate the Lebesgue Density Theorem. Question: Does every non-locally compact metric space ...
- 6,541
3
votes
1
answer
167
views
Definition of homogeneous or quasi-uniform or almost uniform measure
Let us call a measure $\Lambda$ homogeneous if there is an $\epsilon>0$ so that for all $r>0$ and $x,y$ in the support of $\Lambda$, we have
$$\Lambda(B(x,r))>\epsilon\Lambda(B(y,r))$$
...
- 141
6
votes
1
answer
838
views
Can the projection onto a compact set always be taken to be measurable?
This may be a very basic question.
Let $X$ be a complete metric space and let $T$ be a compact subset of $X$. Say that a function $\pi: X \to T$ is a projection if
$$
d(x, \pi(x)) = d(x, T) \quad \...
- 370
7
votes
1
answer
1k
views
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
2
votes
1
answer
1k
views
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 (...
- 6,541