# Questions tagged [geometric-probability]

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193
questions

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### Measure estimates of $\delta$-neighbourhood of compact sets

I am interested in the estimating from above the measure of a compact set $K$ by a sequence of sets $K_n$, converging to it in the Hausdorff metric. As such I am looking for known conditions that give ...

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1
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126
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### 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}\...

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45
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### A sufficient condition for the decomposition of a bounded random vector

Given a matrix $A \in \mathbb{R}_{+}^{n\times m}$, where $n<m$ and rank($A$)=n. Define a set (a zonotope) $\mathcal{C}(A)=\{(x_1,x_2,\ldots,x_n)|\sum_{i=1}^m{\bf{a}}_ix_i,x_i \in [-1,1]\}$, where ${...

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3
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294
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### Under what general conditions is the set $S := \left\{\int_{X}v(x)\pi(x)\,\mathrm{d}P(x) \mid \pi: X \to A\right\}$ closed?

Let $X$ be a compact subset of $\mathbb R^n$ and let $A$ be a compact subset of $\mathbb R^k$. Let $P$ be a probability distribution on $X$ and $v$ be a $P$-measurable function from $X$ to $\mathbb R^{...

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52
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### Upper-bound for bracketing number in terms of VC-dimension

Let $P$ be a probability distribution on a measurable space $\mathcal X$ (e.g;, some euclidean $\mathbb R^m$) and let $F$ be a class of funciton $f:\mathcal X \to \mathbb R$. Given, $f_1,f_2 \in F$, ...

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100
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### Given p1=P(X1>X2), p2=P(X1=X2) and p3=P(X1<X2) (with necessarily p1+p2+p3=1), what is λ1 and λ2？ [closed]

In the following three equations : p1, p2, p3 are known, find λ1=? , λ2=?
(with necessarily p1+p2+p3=1)
$$ p_1 = \sum_{0 \leq k_2 \lt k_1 \in Z}(\frac {\lambda _1^{k_1}} {k_1!}\cdot{e^{-\lambda _1}})(\...

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1
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62
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### 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) ...

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150
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### A question about average deviation of given $n$ complex numbers

This question just came to my mind and I have no idea as to how to approach it. Let $z_1,z_2,\dots,z_n$ be $n$ be any complex numbers in the unit disc $|z| \leq 1.$ Consider a complex function on the ...

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170
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### For $x_1,...,x_n$ iid random on sphere of radius $\sqrt{d}$ in $R^d$, what is a good upper-bound on min distance of $x_{n}$ from the other $x_i$'s?

Let $n$ and $d$ be large positive integers with $n \le d^\gamma$, for some absolute constant $\gamma>0$; i.e., $n$ is at most polynomial in $d$. Let $x_1,\ldots,x_n,x_{n+1}$ be drawn iid from the ...

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209
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### Malliavin calculus and geometric interpretation of $\nabla \cdot ({\nabla F(x)}{\|\nabla F(x)\|^{-2}})$, with regards to the surface $S = \{F = 0\}$

Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define
$$
\alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(...

3
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1
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288
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### Defining a probability distribution on each tangent space of a manifold?

I've been reading about probability on manifolds. What bothers me is that there's no clear way to generalize the Gaussian distribution to manifolds. If we instead assign a probability distribution to ...

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1
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110
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### Tightly placed sphere caps

Problem. Fix a positive real $r>0$ and a positive integer $n>0$. Consider an independent, identical sample $X_1, \ldots X_n$ drawn from the uniform distribution over the unit $d$-sphere $\mathbb ...

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161
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### Time-derivative of integral over sub-level set $s(t) := \int_{f^{-1}((-\infty,t])}p(x)dx$

Let $\mu$ be a probability distribution on $\mathbb R^d$ with "sufficiently regular" density $p$. Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently regular" function. Finally, ...

2
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1
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83
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### Examples of "almost" Ahlfors regular measures

Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ such that there are $c,C,d,D>0$ satisfying: for every $x \in \mathbb{R}^n$ and every $r>0$
$$
c r^d \leq \mu(B(x,r)) \leq Cr^D.
$$
Let'...

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1
answer

194
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### Is $\int_{-c}^c |A \cap (x + A)|\, dx$ maximized when the measurable subset $A \subseteq \mathbb R$ is an interval centered at the origin?

Let $A$ be a nonempty measurable subset of $\mathbb R$, with Lebesgue measure $|A|=1$, and let $c>0$. Define the scalar $I(A)$ by
$$
I(A) := \int_{-c}^c |A \cap (x + A)|\, dx,
$$
where $x+A := \{x +...

2
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1
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101
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### Existence of preferred direction for a random vector with arbitrary distribution on sphere, under a condition on its covariance matrix

Let $X$ be random vector on the unit-sphere $S_{n-1}$ in $\mathbb R^n$. We don't assume that the distribution of $X$ is uniform on $S_{n-1}$
I'm interested in proving the existence of a (...

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24
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### Bound on distortion of norm of high-dim random variable upon projecting onto eigenspace spanned top $k$ eigenvectors of empirical covariance matrix

Let $X$ be a random vector in $\mathbb R^d$ with some "nice distribution" (with a covariance matrix, etc.). For example, for concreteness, one may assume that $X \sim N(0,\Sigma)$ for some $...

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131
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### $\newcommand\v{\operatorname{vol}_d(C}$Compact subsets of $ℝ^d$ which maximize $\inf_{|v|\le1}\dfrac{\v\cap(𝜀v+C))}{\v)}$ for fixed $\v)$ and $𝜀>0$

Let $\operatorname{vol}_d$ be the volume measure on $\mathbb R^d$ and let $B_d$ be the unit-ball. For $\varepsilon \ge 0$ and a compact subset $C$ of $\mathbb R^d$ with $\operatorname{vol}_d(C)>0$, ...

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2
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134
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### Building the Wasserstein space by pushforwards

Let $\mathbb{R}^d$ denote the $d$-dimensional Euclidean space, $\mathcal{W}_2(\mathbb{R}^d)$ denote the $2$-Wasserstein space with respect to the $d$-dimensional Euclidean space $\mathbb{R}^d$. Let $...

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1
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68
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### On the Lipschitz continuity of the unit-normal vector field of a polytope

Let $\mu$ be a probability measure on $\mathbb R^n$ and let $P$ be a compact polytope in $\mathbb R^n$. For any $x \in \mathbb R^n \setminus P$, let $p(x) \in P$ be (unique!) point in $P$ which is ...

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194
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### Comparing two multivariate normal distribution

Let $\mathbf{Z}\sim N(\boldsymbol{\mu},\mathrm{\Sigma})$, where
\begin{equation}\label{Eq.Mean}
\boldsymbol{\mu}^{\rm T} = \delta[-\sqrt{\frac{x_1x_2}{x_1+x_2}},\frac{-1}2\sqrt{\frac{x_1x_3}{x_1+x_3}},...

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### Concentration of number of vertices in convex hull of random points

Dwyer showed that given a set of $n$ points $P$ which were uniformly and independently sampled from the unit cube $[0,1]^d$, if $X$ is the number of vertices in the convex hull of $P$, then
$$\mathbb{...

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### Optimal distribution on $k$-dimensional subspaces of $R^n$ which maximizes $\mathbb E_{V \sim D}\|Proj_V(x)\|^2$, given side informaiton on fixed $x$

Let $d$ be a large integer and let $ k \in [1,d]$ be another integer. Let $V$ be uniform over the grassmannian $G_{k,d}$ of $k$-dimensional subspaces of $\mathbb R^n$, and let $P_V:\mathbb R^n \to \...

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1
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145
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### When are Wasserstein spaces $CAT(\kappa)$?

Let $(X,d)$ be a complete and separable metric space and, for $1\leq p<\infty$, let $(\mathcal{P}_p(X,d),W_p)$ be the $p$-Wasserstein space on $(X,d)$. For which $p$ and $(X,d)$ is $(\mathcal{P}_p(...

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378
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### Isoperimetric inequality for $\epsilon$-expansion of a set only along a certain subspace

Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = ...

3
votes

1
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116
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### Regularity of transport map

Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with first moment and suppose that both $\mu$ and $\nu$ have a densities with respect to the $n$-dimensional Lebesgue measure. Fix some ...

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88
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### What is the limiting marginal distribution of a fixed number of coordinates of a random point drawn uniformly on large-dimensional sphere?

Let $X=(X_1,\ldots,X_d)$ be uniformly-distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. It is well-known that in the limit $d \to \infty$, the marginal distribution of $X_1$ converges ...

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### Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere

This question is related to: https://math.stackexchange.com/q/4270522/168758
Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...

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290
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### The infinity Wasserstein distance $W_\infty$ and the weak topology

Let $X$ be a compact metric space (without isolated points). The $\infty$-Wasserstein distance $W_\infty$ on the space of Borel probability measures on $X$ can be described as $$W_\infty(\mu,\nu) = \...

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163
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### Concentration of $k$-th pairwise distance of random points in a unit square

For $1\leq i \leq n$, let $X_i\sim \text{Uniform}(0,1)$, $Y_i \sim \text{Uniform}(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k$-th smallest pairwise distances across the $n$ ...

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2
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324
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### Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$

Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$.
Question.
What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...

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1
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88
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### Norm contrained Gaussian distribution

Let $X$ be a multivariate normal $\mathcal{N}(\mu, \Sigma^2)$ and let $X$ be anisotropic, that is I am considering $\Sigma$ to be a diagonal matrix but the elements on the diagonal might be different.
...

6
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275
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### Almost evenly distributed spherical random vectors

Consider $n$ i.i.d spherically distributed random vectors $z_1 ,\cdots , z_n \sim \text{Unif}(\mathbb{S}^{d-1})$. What is the best lower bound on $n$ for which whp there exists a constant $c>0$ ...

6
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660
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### Expected absolute value of the average of two points from the disc

Looking at Average distance of the mean of n random complex numbers in a unit disc, I tried to figure out
what is the expected absolute value $|\frac{z_1 + z_2}{2}|$ of two numbers $z_1, z_2\in\...

6
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2
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361
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### 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 ...

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### Comparison inequality between Sobolev-seminorm w.r.t spherical uniform distribution and gaussian distribution

Let $d$ be a large positive integer. Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function. Let $X$ be uniformly distributed on the unit-sphere $S_{d-1} := \{x \in \mathbb R^d \|...

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### Geometric of sphere caps with infinite radius $R$ (asymptotic) in dimension $C$

Let $B_C(R)$ be the Ball in $R^C$ with radius $R$, the vectors $w_1,w_2,\dots,w_{C-1}\in R^C$ is the normal vector of hyper half plane, the area
$$\mathcal{K}=\{x:w_1^Tx\leq0, w_2^Tx\leq0,\dots,w_{C-1}...

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115
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### Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere

Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...

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196
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### Bounding Brownian motion and an Ito process simultaneously

Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...

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142
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### Almost independence of $x^\top a$ and $x^\top b$ for $x$ uniform on the sphere in $\mathbb R^d$ and $a,b \in \mathbb R^d$ with $a^\top b = 0$

Let $d$ be a large positive integer. Let $x$ be uniformly distributed on the unit-sphere in $\mathbb R^d$ and let $a$ and $b$ be perpendicular vectors in $\mathbb R^d$, i.e such that $a^\top b=0$. Let ...

2
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0
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70
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### Local variation of normals on Riemannian submanifolds of $\mathbb R^D$

Let $M^d\subseteq\mathbb R^D$ be a compact manifold, $p\in M$ and $\nu\in N_p(M)$. Suppose that all the principal curvatures in direction $\nu$ are in $[a,b]$ for some $0<a<b<\infty$. Suppose ...

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### Lebesgue measure of a circular arc on the surface of the hypersphere [closed]

The following sub-problem came up in one of my research works:
Suppose that $U$ is a uniform random variable on the surface of the $d$-dimensional sphere with center at the origin and of radius $r$. ...

3
votes

1
answer

170
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### 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 $H$...

5
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1
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168
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### Probability of gaps between coordinates of a random point on the sphere

Let $X=(X_1,\ldots,X_n)$ be a point chosen uniformly at random from the sphere $S^{n-1}\subseteq \mathbb R^n$. Given $a>0$, what is the probability that $|X_1|^2-|X_i|^2\geq a$ for all $i>1$? Is ...

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97
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### 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 ...

3
votes

1
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173
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### 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 ...

28
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954
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### Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments

It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...

2
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0
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117
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### 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
vote

1
answer

81
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### 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 $\...

4
votes

1
answer

184
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### Does there exist a scale invariant random packing of circles in the plane?

I want to construct a scale invariant random packing of the plane with circles.
Here is a way to construct a rotationally invariant, but not scale invariant random packing of the plane with circles:
...