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0
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2answers
89 views

Distribution of the individual coordinates of a uniform random vector on a high-dimensional sphere

Let $X=(X_1,\ldots,X_n)$ be a random vector uniformly distributed on the $n$-dimensional sphere of radius $R > 0$. Intuitively, i think that for large $p$ every coordinate $X_i$ is normally ...
1
vote
1answer
51 views

gaussian isoperimetric result for minimal measure under translation

Consider two spherical Gaussian distributions in $\mathbb{R}^n$, $A = \mathcal{N}(x, I)$ and $B=\mathcal{N}(y, I)$ where the difference in means is $\delta = y - x$. Let $S \subset \mathbb{R}^n$ be a ...
1
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0answers
49 views

Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...
2
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0answers
81 views

Covering a sphere with ellipsoid-products in high dimension

For $\Sigma\geq 0$ a $k\times k$ matrix and large $n$, fix $E:= \{(x_i)_{i=1}^n: \sum_i x_i^\dagger \Sigma x_i \leq n\}$. Fix $(z_m)_m$ as $M$ points iid uniform on $\mathbb{S}^{nk-1}\subset \mathbb{...
0
votes
1answer
114 views

Does log-concave approximable distribution satisfy transportation-cost inequality?

Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition $$ \...
1
vote
1answer
128 views

Hausdorff distance is a lower (or upper bound) for what probability metric?

In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that $$ d(A, B) \le W(\mu|_A, \mu|_B), $$ where $d(A, B):= \...
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0answers
93 views

Is there a precise relationship between ``Geometric Functional Analysis" and high-dimensional probability/information theory?

The 2009 course on GFA by Roman Vershynin (https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf) introduced the subject with this line on the course page, "...
4
votes
1answer
131 views

General distributions with the “transportation-cost inequality” property to piece log-concave distributions

It is now known [Otto et Villani 2000; Cordero et al 2006; etc.] that on an $n$-dimensional smooth Riemannian manifold $X$ and a probability measure $\mu$ on $X$ with density $d\mu \propto e^{-V}dvol$ ...
4
votes
1answer
101 views

What is the nearest-neighbor distribution in this picture?

Consider the following process: sample $n$ points uniformly at random in the unit square, and for each point $i$, let $d_i$ be the distance from $i$ to its nearest neighbor. Finally, let $z_i = d_i\...
4
votes
1answer
78 views

Distribution of the $pn$ shortest edges out of $n$ uniform points, $p\to 0$

Suppose I sample $n$ points independently and uniformly at random in the unit square, and then I select the $pn$ shortest edges between all pairs of points, for fixed $0<p<1$. For large $n$ and ...
1
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0answers
56 views

Upper bound for $\sup_{W_d(Q, P) \le \epsilon} Q(A)$, where $W_d$ is the Wasserstein metric

Let $X=(X,d)$ be a metric space and let $W_d$ denote the Wasserstein metric induced by this metric, on the space of probability distributions on $X$. Let $\epsilon \ge 0$, $A$ be a Borel subset of $X$...
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0answers
31 views

General conditions for Gaussian isoperimetric inequaliteis

Context I'm doing some work in which i need to show that the blow-up $B_\epsilon$ of a Borel set $B$ has large measure for $\epsilon$ sufficiently large. I've found a very attractive tool for doing ...
2
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0answers
32 views

Extreme points in $d$-dimensional quadrant hull of $n$ random points in a halfplane

Finding the asymptotic growth of the expected number of extreme points of the quadrant hull in $d$ dimensions seems to be a well studied problem. Interestingly, the solution appears to vary widely by ...
5
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1answer
232 views

Approximation of Wasserstein distance between $p_\theta$ and $p_{\theta + d\theta}$

Given a parametric family of distributions $\{p_\theta\mid\theta \in \Theta\}$, one can show that under some regularity conditions, the following approximation is valid $$\operatorname{KL}(p_\theta\...
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0answers
28 views

Interpretation of this sum for concave bodies

For some polyhedron, $P$, define $$H(P)=\sum_{e\in E} L_e(\pi - \delta_e)/(4\pi)$$ Where $E$ is the set of all edges of the polyhedron, $L_e$ is the length of edge $e$ and $\delta_e$ is the interior ...
2
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2answers
101 views

Does this formula for caliper diameter hold for concave polyhedra?

I recently asked on MathOverflow and also asked several people I know to prove the following: How do I prove that the average caliper diameter of the polyhedron across all possible rotations is ...
25
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5answers
1k views

Moments of area of random triangle inscribed in a circle

The $2m$th moment of the (random) area of the triangle whose vertices are three independent, uniformly distributed random points on the unit circle appears to be $((3m)!/(m!)^3)/16^m$. Can anyone ...
0
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0answers
135 views

Probability that the perturbed convex hull is larger than the original one

I am wondering if any convex geometers/probabilists have looked at the following question: Given $n$ randomly distributed (not sure what assumption to put there) points in $\mathbb{R}^d$, for each ...
3
votes
0answers
106 views

What is the shape of a random convex polygon on the sphere?

Say $\mathbb{R}^n$ is divided into sectors by $k>n$ random hyperplanes (each hyperplane chosen as the orthogonal complement of a vector uniform on the unit sphere in $\mathbb{R}^n$). Each sector (...
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0answers
83 views

What is probability distribution function or probability density function about radii of two points in a circle

Thank you for reading this question. Thank you for helping me to solve this question. If you cannot give the details, giving the relevant clue(web links) or references is OK as well. Pre-condition:...
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0answers
64 views

Measure of the boundary of the support of a certain function defined by an expectation

Suppose: $\mathcal{S} = \{ S \in \mathbb{R}^d \ | \ S_i > 0, \forall i = 1,...,d \} $ $R$ is a random vector (on some probability space, $\Omega$) such that, $R: \Omega \to \mathcal{S}$. $h : ...
9
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2answers
207 views

Density of a saturated random packing of congruent circles

The problem of the expected density of a saturated random packing of unit circles in the plane can be described as follows. In a circular region $C$ of a large radius pick a point at random and draw ...
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0answers
93 views

Expected Number of Triangles

A unit square is divided up with $n$ random lines. The random lines are chosen as follows, we choose one side of the square and pick a random point on that side. From there we choose a random point on ...
4
votes
3answers
178 views

The distribution of the number of chord intersections

This is a follow-up to this question: Given $n$ random chords of a circle, what is the distribution of the number of intersections? Random is defined by "endpoints uniform on the circle". Update ...
3
votes
2answers
284 views

The Largest Piece of Circumference

We add $n$ random lines to the unit disc. We do so by adding two points on the disc, transcribing a line between them and extending that line to the boundary of the disc, namely the unit circle. Each ...
3
votes
0answers
80 views

Expected Largest Area with Random Lines

We take the unit square and have it divided by $n$ lines which are chosen randomly. We choose the lines as follows, choose one of the four sides of the square at random and then choose a random point ...
3
votes
2answers
104 views

Concentration of norm of projection onto a subspace

Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. Let $V$ be a linear subspace of dimension $k$ and let $P_V(x)$ be the orthogonal projection of $x$ onto $V$. I ...
1
vote
0answers
93 views

Expected Area of Randomly Made Triangle [closed]

Say we have a piece of length one, and then we draw twice from a bin of sticks in which there are an infinite amount of sticks with lengths evenly distributed on $[0,1]$. In cases where a triangle can ...
0
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0answers
57 views

Expected uninterferred with perimeter on square with dots

We play a game on a square in which we randomly choose a point on the perimeter of the square and place a dot which occupies zero area there. We do this for $n$ turns. What is the expected length of ...
2
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0answers
188 views

norm of projection of a random vector on the sphere onto a linear subspace

Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{D-1}$ and let $\mathcal{V}$ be a linear subspace of dimension $m$. Then it is known that the euclidean norm of the ...
0
votes
0answers
48 views

Probability that density at point p and density at point near p are similar in high dimensional gaussian

Let's consider a Gaussian with $d>>1$, $N(x)$, and sample a point $p$ from it. Let $S(p,r)$ be the hypersphere centered at p of radius r. Now for any probability $f$, let $r_f(p)$ be the radius ...
8
votes
2answers
241 views

What is the probability that these sets intersect?

Let $A$ be the subset of $\mathbb{R}^n$ defined by $A=\{x\in\mathbb{R}^{n}:|x_{1}-x_{n}|+\sum_{i=1}^{n-1}|x_{i+1}-x_{i}|\leq d\}$ for a given $d$. Next, sample a point $p$ uniformly in the unit cube, ...
1
vote
1answer
129 views

When is the second largest Gaussian r.v. the largest in the stochastic sense?

Let $X_1, \ldots, X_n$ be jointly Gaussian, each of which is marginally distributed as a standard Gaussian $N(0,1)$. It is well known that $\max |X_i|$ achieves the maximum in the stochastic sense if $...
4
votes
1answer
170 views

Random spherical caps cover a spherical cap

Let $S^{n-1}$ be the unit sphere in $n$ dimensional Euclidean space. Define the spherical cap at $x \in S^{n-1}$ with angle $\theta$ to be $C(x,\theta) = \{z \in S^{n-1} \mid z^\top x \geq \cos(\theta)...
4
votes
1answer
84 views

Union of random half spaces cover a ray

Let $x, y \in \mathbb{R}^{n}$ be two fixed unit vectors with angle $\alpha \in (\frac{\pi}{2}, \frac{3\pi}{4})$. Define the positive half space associated with a vector $z$ to be $\mathcal{H}(z) = \{h ...
3
votes
1answer
297 views

Uniformly Random Independent Unit Vectors Inner Product Limit

Suppose $V$ is a $N\times n$ matrix the columns of which are independently distributed uniformly on $\mathbf S^{N-1}$ the surface of the unit sphere in $\mathbf R^N$. I conjecture that $V^TV$ ...
4
votes
2answers
437 views

Total progeny of a Galton-Watson branching process - standard textbook question

While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly. ...
1
vote
1answer
151 views

Properties of moment generating function of random walk on unit sphere

Question in brief Let $a$ and $b$ be unit vectors in $\mathbb{R}^d$. Let $f$ be the $1-step$ transition function of a random walk on the $d$ dimensional unit sphere. I am interested in evaluating $\...
5
votes
1answer
420 views

The expectation of two sides of rectangle is equal. Can we deduce that in the expectation the rectangle is not very far from being a square?

Let $T$ be a set of $n\ge 3$ points in the plane such that not all of them lie in a common line. Pick two distinct points $\{a=\left( \begin{array}{c} a_{1} \\a_{2} \end{array} \right) ,b=\left( \...
3
votes
0answers
87 views

Partitioning a sphere with a random tessellation

I have a $d$-dimensional sphere of radius $1$. I now tessellate the space with $d$-dimensional cubes of side $L$ and uniform distribution of the origin of the tessellation. Thus the size of the ...
5
votes
1answer
133 views

Does there exist a Penalized Conditional Expectation?

In my recent work I've become interested in working with the minimizer of $$ \mathbb{E}[(Y-Z)^2] + \lambda P(Z), $$ $Y$ is an observed random variable, $P$ is a positive-convex penalty function, $Z$ ...
1
vote
2answers
593 views

Minimum distance between $n$ points in a cube

What is the expected value for the minimum distance between $n$ points placed randomly, assuming a uniform distribution, within a cube of volume $V$?
4
votes
0answers
147 views

Volume difference in random approximation of polytope

The following easily stated problem has arisen in my research. However, it's outside my field, and I'm unfamiliar with the literature. I would greatly appreciate any references. Let $K\subset \...
13
votes
4answers
433 views

Alignment of random points

Whenever I draw randomly about ten points, I see that there will be always 3 points that are "almost" collinear. This observation leads me to considering the following questions: Question 1: Suppose $...
9
votes
2answers
465 views

Minimum separation among $m$ random points on an $n$-dimensional unit sphere

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{...
12
votes
2answers
277 views

Shortest path through $n^{1/3}$ points out of $n$

Say I sample $n$ points uniformly at random in the unit cube in $\mathbb{R}^3$, and then I look for the shortest path through $n^{1/3}$ of those points (rounding up, say). What happens to the length ...
3
votes
0answers
110 views

Algorithm to calculate moments of uniform distribution on convex polyhedra

There is system of linear inequalities $$ Ax \leq K, $$ $$ x\geq a, x\leq b. $$ $A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$. Suppose that on set of solutions ...
4
votes
1answer
209 views

Upper tail concentration of sample covariance matrices

I'm interested in concentration of the following random matrix sum in spectral norm $\frac{1}{m}\sum_{k=1}^m b_k^2\mathbf{a}_k\mathbf{a}_k^*$ Here $\mathbf{a}_k\in\mathbb{R}^n$ are i.i.d. standard ...
8
votes
1answer
351 views

Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^d$ ...
6
votes
3answers
244 views

Probability of random geodesics on the half-sphere intersecting

4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim half-sphere. Here, uniform means that, probability that a point lies on a surface ...