All Questions
Tagged with mg.metric-geometry pr.probability
223 questions
4
votes
0
answers
52
views
Isomorphism of Wasserstein space implies isomorphism of base spaces?
Assume $(X_i,d_i)$ are polish spaces (or compact metric spaces) for $i=1,2$.
Further assume that the 1- Wasserstein spaces $(P_1(X_1),W_1)$ and $(P_1(X_2),W_1)$ are isometrically isomorphic. Does that ...
9
votes
0
answers
240
views
Does there exist such a probability distribution?
Does there exist a probability distribution over the set $\{(x,y,z)\in[0,1]^3\colon x+y+z=3/2\}$ whose projection on each of the three coordinate axes is the uniform distribution over the interval $[0,...
- 128k
0
votes
0
answers
24
views
Is there a log-concave distribution not spherical symmetric s.t $ \langle X, \theta \rangle$ is almost normal for all directions $\theta$?
Klartag's results indicate that for a log-concave isotropic random vector, with high probability over $\theta$, $\langle X, \theta \rangle$ is close to a normal distribution.
It is known that for the ...
- 11
4
votes
0
answers
87
views
Statistics of random Voronoi S-tessellations
Given a locally finite set of points $\{x_1,x_2,\dots\}\subset\mathbb{R}^d$, the Voronoi cell of a point $x_{i}$, denoted by $C(x_{i})$, consists of all the points in $\mathbb{R}^d$ that are closer to ...
- 41
2
votes
0
answers
100
views
Distributions of random walks on boundaries of balls in hyperbolic metric spaces
Suppose $G$ is a finitely-generated non-elementary hyperbolic group and consider a symmetric random walk on the Cayley graph $\text{Cay}(G,S)$ with generating set $S$. Denote the set of points $B_{\...
- 21
2
votes
1
answer
81
views
Rate of convergence of random samples wrt Hausdorff distance
Let $X$ be a compact metric space with a probability measure $\mu$. We can draw random samples $X_n = \{x_1,\cdots, x_n\}$ from $X$ using $\mu$, and I am interested in the rate of convergence of $X_n$ ...
- 305
9
votes
2
answers
658
views
Probability that randomly chosen balls have a nonempty common intersection
Fix some $0 < r < 1$. A collection of points $x_1, \dots, x_n$ are chosen independently and uniformly at random from the closed unit ball in $\mathbb R^d$.
What is the probability that the ...
- 6,155
2
votes
0
answers
124
views
Generalization of the triangle inequality to complex exponents: What is $P\left(\left| x^{a+bi} + y^{a+bi} \right| \ge \left|z^{a+bi}\right|\right)$?
Let $x \le y \le z$ be the length of the sides of a triangle whose vertices are uniformly random on the circumference of a circle. In this question, it was proved that if $a \ge 1$, then the ...
- 5,470
13
votes
1
answer
484
views
A probability involving areas in a random pentagram inscribed in a circle: Is it really just $\frac12$?
This question was posted at MSE but was not answered.
The vertices of a pentagram are five uniformly random points on a circle. The areas of three consecutive triangular "petals" are $a,b,c$...
- 3,527
0
votes
0
answers
77
views
Wasserstein space isomorphic to original space?
Is there a complete measurable metric space $(X,d)$ for which its $p$-Wasserstein space $W(X)$ is isometrically isomorphic to $(X,d)$ for some $p \in [1,\infty]$?
Note that there is a canonical non-...
6
votes
0
answers
197
views
What are compact manifolds such that GROWTH (of spheres volumes) is well approximated by the Gaussian normal distribution?
Consider some compact Riemannian manifold $M$. Fix some point $p$.
Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$.
Consider growth function $g(r)$ to be ...
- 24.9k
3
votes
1
answer
156
views
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
2
votes
0
answers
64
views
Limiting distribution of separated points in a unit square
Let $n$ and $r$ be fixed, and consider the following process, with $S=\emptyset$ to start:
For $i\in\{1,\dots,n\}$:
Sample a random point $X$ in the unit square.
If $X$ is a distance at least $r$ ...
- 4,049
3
votes
0
answers
228
views
Are 1-Wasserstein and 2-Wasserstein distances between multivariate normal distributions equivalent?
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$W^p_p(\nu_{1},\nu_{2})=\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,dy)$...
- 1,048
5
votes
1
answer
138
views
Complexity and length
Suppose we define continuous piecewise linear functions $f$ on $[0,1]$ using your favorite programming language, or by finite automata, or by any other suitable machine. Define the complexity $H(f)$ ...
- 2,934
15
votes
0
answers
398
views
Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?
On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...
- 3,527
5
votes
0
answers
184
views
Question about $n$ random points in a regular polygon, and a limiting probability
Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is ...
- 3,527
7
votes
1
answer
508
views
An order statistics problem with some interesting geometry
Let $a_n$ be a given sequence of positive numbers, and $X_n$ a sequence of independent random variables with each $X_n$ uniformly distributed on $[0, a_n]$.
Question: Let $N \geq 2$ be an arbitrary ...
- 6,155
6
votes
1
answer
180
views
Expected value of the length of the shortest non-zero vector in a lattice?
$\DeclareMathOperator\SL{SL}$What is the expected value of the length of the shortest non-zero vector in a (unimodular) lattice? I.e., let $G=\SL_n(\mathbb{R})$ with Haar measure $\mu$, $\Gamma=\SL_n(...
- 609
1
vote
0
answers
83
views
Closed form volumes for intersecting modified cylinders
This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the ...
- 383
0
votes
0
answers
79
views
Geometry of inner products between the unit vector and several given vectors
Let $\mathcal{S}$ denote the set of all unit complex-valued $d$-dimensional vectors, i.e.,
$$
\mathcal{S} \triangleq \left\{ \mathbf{s}\in \mathbb{C}^{d} \mid \mathbf{s}^{\mathrm{H}}\mathbf{s}=1 \...
- 550
3
votes
0
answers
187
views
Approximating any $d$-dimensional convex shape that occupies a constant fraction of its bounding box with a polytope having $\mathrm{poly}(d)$ facets
Given any convex set $A\in\mathbb{R}^d$, we denote by $V(A)$ its $d$-volume. Furthermore, given any two convex sets $A_1,A_2\in\mathbb{R}^d$, we denote by $V_{A_1,A_2}$ the $d$-volume of the symmetric ...
- 1,677
7
votes
0
answers
162
views
Approximating any convex shape in $\mathbb{R}^d$ with a polytope having $\mathrm{poly}(d)$ facets
We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...
- 1,677
2
votes
1
answer
190
views
Estimating the volume of a convex shape in higher dimensions based only on normal sections
We are given a $d$-dimensional convex shape $S$ inscribed in the hypercube $[-1,1]^d$. We want find an approximation of its volume based only on a set of curves given by the intersection of the $S$ ...
- 1,677
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
3
votes
1
answer
218
views
Bounding the number of facets of a polytope to approximate a given convex shape in higher dimensions
We are given a convex shape $S$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume $V(S)$ of $S$ be $\tfrac12$ (I guess nothing changes for any other fixed ...
- 1,677
4
votes
0
answers
144
views
Approximation of a convex shape in the $d$-dimensional Euclidean space for $d\gg 1$
We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant ...
- 1,677
7
votes
1
answer
186
views
$d$-ball approximation for $d\gg 1$ with a convex hull of random points on its boundary
Given a $d$-ball $\mathcal{S}^{d}$, let $P_n$ a set of $n$ points selected uniformly at random on the boundary $\mathcal{S}^{d-1}$ of $\mathcal{S}^{d}$. Let $\mathcal{C}_n$ the convex hull of $P_n$. ...
- 1,677
11
votes
1
answer
323
views
Probability distribution for the number of triangles containing the center of a circle
Pick $n$ points randomly on a circle centered at the origin. Let $X$ be the number of the ${n \choose 3}$ triangles with those vertices that contain the origin in their interior. For fixed $n$, what ...
- 506
0
votes
1
answer
247
views
Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions
We are given $\mathbf{p}\in\mathbb{R}^d$, where $d\gg 1$. Let $\mathbf{v}$ be a point selected uniformly at random from the unit $(d-1)$-sphere $\mathcal{S}^{d-1}$ centered at the origin $\mathbf{0}\...
- 1,677
1
vote
1
answer
288
views
Probability that three vectors of a unit sphere lie on one side of a hyperplane if angle between the vectors are given
As the title says, How to find the probability of vectors a, b, c, on some unit sphere, all lies on same side of some hyperplane passing through the origin. Information present are the angles between ...
- 23
5
votes
0
answers
74
views
Concentration bound on additive functions with constraints
Given a family of sets $F \subseteq P(\{1,\ldots,n\})$. I define the function $f_F:[0,1]^n \rightarrow R$ to be $f_F(x_1,\ldots,x_n)= \max_{S \in F} \sum_{j \in S} x_j$.
Given a series of independent ...
- 107
3
votes
1
answer
350
views
Talagrand's inequality for L1 norm
I have a series of $n$ independent random variables $X_1,\ldots, X_n$, each with the support $[0,1]$, and a monotone convex function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ that is 1-Lipshitz in L1 ...
- 107
1
vote
1
answer
115
views
Approximating the probability of a half-space using random Voronoi diagrams
Fix a half-space $H = \{x_1 \geq 0: ~ (x_1,\dots,x_n) \in \mathbb{R}^n\}$. Let $p$ be a distribution with support in $\mathbb{R}^n$. I am interested in the following way of estimating the weight $p(H) ...
- 33
1
vote
0
answers
153
views
Is there a polynomial expression for the volume of the following set?
Denote the unit $\ell_2$ ball in $\mathbb{R}^n$ as $\mathcal{B}_n$. It is widely kown that for a convex body $\mathcal{K}\subseteq \mathbb{R}^n$, the $n$-dimensional volume of the parallel body $\...
- 550
2
votes
2
answers
248
views
On an angle distribution of a random linear subspace of a given dimension
$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
- 128k
1
vote
1
answer
100
views
$L^p$-barycenters via continuous selectors
Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following ...
- 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&...
0
votes
1
answer
146
views
Upper bound of Wasserstein distance given by subvariables of codim 1
recently I am considering the upper-bound of Wasserstain distance. Say we have random vectors $X,Y$ of dimension $n$, and let $\tilde{X}_i (\tilde{Y}_i,$ resp.) be the $(n-1)$-dim random vector of $X (...
- 363
6
votes
2
answers
497
views
Average distance of the mean of $n$ random complex numbers in a unit disc
Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers distributed uniformly and randomly over the unit disc $x^2+y^2 \leq 1$. Let $z$ be the complex number defined by the mean of the of these numbers,that ...
- 826
2
votes
1
answer
245
views
Union bound over infinitely many events
Let $p_1, ... ,p_n$ be chosen independently from the uniform distribution on the unit torus $[0,1]^2$.
I want to prove a theorem of the form: "With high probability, every circle of radius $r$ ...
- 1,633
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
4
votes
1
answer
567
views
Random graphs and Benjamini-Schramm convergence
I am looking for literature on the question whether a randomly chosen sequence of $k$-regular graphs converges in the Benjamini-Schramm sense to the universal covering with probability one.
There are ...
user130903
3
votes
1
answer
206
views
Random planes separating points in $\mathbb{R}^3$
We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$...
- 1,677
2
votes
1
answer
151
views
Given an input point in $\mathbb{R}^n$, select (one of) the closest point(s) from a fixed large set of points given in advance
We are given a set $S$ of $m\gg 1$ points in $\mathbb{R}^n$.
In the problem I am trying to solve, in a sequential fashion, we obtain a new point $p_r\not\in S$ at each round $r\ge 1$ and the goal is ...
- 1,677
1
vote
1
answer
117
views
Modulus of continuity of parameterizing Wasserstein
Let $x_1,\dots,x_n\in X$ some Polish space $X$ and let $\Delta$ be the probability simplex in $\mathbb{R}^n$. Consider the map sending every $(w_1,\dots,w_n)\in\Delta$ to the finitely supported ...
0
votes
0
answers
113
views
How much a probability distribution is non-uniform in a convex subspace of $\mathbb{R}^d$?
I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another.
Could you please ...
- 1,677
3
votes
1
answer
321
views
Is disintegration continuous?
Let $X,Y$ be Polish spaces and suppose that $X$ is compact. Denote by $\mathcal{Mes}(X,\mathcal{P}(X\times Y))$ the set of (Borel) measurable functions from $X$ to the set of Borel probability ...
- 5,405
2
votes
0
answers
174
views
Random sets of points and hyperplanes in high dimensions
We are given a set $X$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \in\mathbb{R}^d$ selected uniformly at random from the unit origin-centered ball $\mathcal{B}^{d}$.
Consider the random ...
- 1,677
1
vote
1
answer
92
views
Geometric sampling problem in the Euclidean space in high dimensions
Let $T$ be the triangle whose vertices are three given points $\mathbf{x}, \mathbf{y}, \mathbf{z}\in\mathbb{R}^d$.
Question: What computationally efficient strategy can we use to sample a point $\...
- 1,677