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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 $a,b,c$...
Dan's user avatar
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13 votes
1 answer
710 views

If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$

Posting this question in MO since it is unanswered in MSE Let $(a,b,c)$ be the side of a triangle. In its most general linear form, the triangle inequality can be expressed as: Does $ax + by \ge c$ ...
Nilotpal Kanti Sinha's user avatar
10 votes
1 answer
657 views

Find the area of the region enclosed by $\sin^p x+\sin^p y=\sin^p(x+y)$, the $x$-axis and the $y$-axis (comes from a probability question)

Consider the graph of $\sin^p x+\sin^p y=\sin^p(x+y)$, where $x$ and $y$ are acute, and $p>1$. Here are examples with, from left to right, $p=1.05,\space 1.25,\space 2,\space 4,\space 100$. Find ...
Dan's user avatar
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13 votes
8 answers
1k views

The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$. There is a convoluted proof that $P(ab>c)=\frac12$. But since the ...
Dan's user avatar
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2 votes
0 answers
59 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 $r$ ...
Tom Solberg's user avatar
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2 votes
2 answers
163 views

Random partition of an interval – Dirichlet distributed?

Let $X_1, \ldots, X_N \sim \operatorname{Unif}[0,1]$ and consider the intervals between successive order statistics: $[0, X_{(1)}], [X_{(1)}, X_{(2)}], \ldots, [X_{(N)}, 1]$. What is the distribution ...
gusl's user avatar
  • 57
13 votes
1 answer
967 views

A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?

A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points. For example, here are $20$ random ...
Dan's user avatar
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15 votes
0 answers
378 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 ...
Dan's user avatar
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2 votes
1 answer
164 views

Small deviations of real log-concave random variable

I am working with a log-concave real random variable, that has a density $f(x) = \exp(-\varphi(x))$ with $\varphi$ convex. Assuming that $X$ is centered and has unit variance ($\mathbb{E}X=0$, $\...
Hugo Ch's user avatar
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2 votes
1 answer
261 views

Does this KL divergence inequality hold?

Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that $$ \frac{y\left( x \right)}{p\left( x \right)}=\frac{\...
Jiacai Liu's user avatar
0 votes
0 answers
39 views

Intersection of subspace of cyclical rotations with orthant

In an $N$-dimensional real Euclidian space, let an orthant be specified by a vector $\underline{x}_0 = \{x_1, x_2, \dots, x_N\}$ where the components $x_k$ are binary in the sense that $x_k = \pm 1$...
Andreas's user avatar
  • 101
0 votes
0 answers
37 views

Random variables and marked point processes

Let $(\Omega,\mathcal{A},\mu)$ be a probability space. Suppose that $f:\Omega\times\mathbb{R}\to \mathbb{R}$ is a given function (that can be continuous in its second entry). Is it possible to ...
Jcnak's user avatar
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2 votes
0 answers
254 views

Why is it impossible to create a numerically balanced die with more than 120 sides?

I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
Matthieu Nauly's user avatar
2 votes
0 answers
138 views

An attempt to define expected value of a Riemannian manifold valued random variable - what'll go wrong?

Let $X:\Omega\to (M,g)$ be a random variable taking values in a Riemannian manifold $(M,g)$ with the Riemannian volume form denoted by $dvol_g(x).$ We know that there's no standard way to generalize ...
Learning math's user avatar
7 votes
1 answer
290 views

Local Lipschitzness of parameterization of Gaussians in Wasserstein space

Fix a positive integer $n$ and consider the $2$-Wasserstein space $\mathcal{P}_2(\mathbb{R}^n)$. Let $X$ be the cone of $n\times n$ symmetric positive semidefinite matrices with Frobenius norm and ...
Justin_other_PhD's user avatar
0 votes
1 answer
109 views

How many samples do you need to get constant dispersion?

Let $C_n$ be the hypercube $[-1,1]^n$. For $a_1,\cdots,a_s \in C_n$, define its dispersion $D(a_1,\cdots,a_s)$ as $\max_{x \in C_n}\min_{i \in [s]} \|x-a_i\|_{2}$. Let $0< \lambda < 1$ be a ...
Mathews Boban's user avatar
4 votes
0 answers
49 views

Quantifying error in the reconstruction of convex polytopes from moments

The problem of reconstructing a geometric object from its moments is of interest in a variety of fields. In the paper The Inverse Moment Problem for Convex Polytopes, the authors show that a convex ...
Lucas Blakeslee's user avatar
2 votes
1 answer
191 views

Dispersion of a "random" subset of $[-1,1]^2$

Let $C$ be the square $[-1,1]^2$. Let $a_1,\dots,a_m$ be points chosen independently and uniformly at random from $C$. Let $d_m$ (dispersion) be the random variable $\max_{x \in C}{\min_{j \in [m]}{\|...
Mathews Boban's user avatar
0 votes
0 answers
106 views

Support function of the intersection of a hyper-ellipsoid and a Euclidean ball

Le $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_d$ be positive numbers. For any $x \in \mathbb R^d$ and $r \ge 0$, define $\gamma(x,r) := \sup_{z \in E(r)}x^\top z$, where $$ E(r) := E \cap B_2^d(r)...
dohmatob's user avatar
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0 votes
1 answer
81 views

Can a measure on a finite metric space be Alhfors regular?

Recall that a probability $\mu$ measure on a metric space $(X,d)$ is called Ahlfors $q$-regular if there are $0<c\le C$ such that: for $\mu$-a.e.\ $x\in X$ one has $$ cr^q \le \mu(B(x,r)) \le Cr^q, ...
Math_Newbie's user avatar
0 votes
1 answer
96 views

Order of orthant probabilities in a prolate multinormal distribution

This is inspired by the negative answer to the conjecture in Which orthant probabilities are the largest? (For a multivariate normal distribution). Suppose $X$ has the $k$-dimensional multivariate ...
Jukka Kohonen's user avatar
1 vote
2 answers
75 views

Variant of the Schläfli problem about random points on the hypersphere

I believe it was Ludwig Schläfli who first worked out the probability $m$ points uniformly distributed on the $n$-sphere all lie in the same hemisphere. In the limit of large $n$ this probability ...
Veit Elser's user avatar
  • 1,065
2 votes
1 answer
77 views

Distribution of scaled Johnson-Lindenstrauss transforms

Suppose that $\mathcal{D}$ is a Johnson-Lindenstrauss (JL) distribution on $\mathbb{R}^{r\times n}$ ($1 \le r \le n$), meaning that there exist constants $\epsilon, \delta \in(0,1)$ such that $$ \...
Nuno's user avatar
  • 213
1 vote
1 answer
202 views

Gaussian width of intersection of cube and ball in high-dimensional euclidean space

Let $d$ be a large positive integer and fix $r \ge 0$. Set $S := B_2^n \cap [-r,r]^d$, where $B_2^d$ is the euclidean unit-ball in $\mathbb R^d$. Finally, let $\omega(S)$ be the Gaussian width of $S$, ...
dohmatob's user avatar
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4 votes
1 answer
102 views

Mean width of a simplex as one edge becomes longer

Consider an edge $e$ of a simplex $C$. If we increase the length of $e$ while keeping the length of other edges constant, the width of $C$ increases in some directions and decreases in other ...
Pluviophile's user avatar
  • 1,514
1 vote
1 answer
88 views

Distance between empirical measures and thickened version

Let $\mathcal{H}$ be a separable Hilbert space and let $x_1,...,x_n$ be points in $\mathcal{H}$. Let $\varepsilon >0 $ be given and consider the measures $$ \mu := \frac1{n}\,\sum_{i=1}^n\, \...
ABIM's user avatar
  • 4,775
1 vote
1 answer
203 views

Integration by parts for indicator of a sphere to indicator of a ball

Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
Nicolas Resch's user avatar
4 votes
1 answer
265 views

Equidistribution of distances of integer points to a circle

I have noticed in the following graph that the euclidean distance between points $k \in\mathbb{Z}^2\cap C_7^1$ ($C_7^1:={}$Circle with radius 7 and shell with thickness 1) and the nearest point on the ...
HyyFly's user avatar
  • 187
0 votes
0 answers
50 views

Inclusion-exclusion in a set of multivariables

I want to determine the risk of a multi-asset portfolio, in a way different from previous attempts. It is because current methods focus on just the correlation coefficient of two variables, while in a ...
R Salimi's user avatar
  • 221
2 votes
0 answers
124 views

Eigenvalues of Witten Laplacian induced by log-concave probability measure on manifold

Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery ...
dohmatob's user avatar
  • 6,764
1 vote
0 answers
82 views

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 ...
Keen-ameteur's user avatar
0 votes
1 answer
239 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}\...
Penelope Benenati's user avatar
1 vote
3 answers
341 views

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^{...
dohmatob's user avatar
  • 6,764
0 votes
0 answers
166 views

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, $f_1,f_2 \in F$, ...
dohmatob's user avatar
  • 6,764
1 vote
0 answers
204 views

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}})(\...
Gang Huizou's user avatar
1 vote
1 answer
89 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) ...
π314's user avatar
  • 33
3 votes
1 answer
178 views

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 ...
AgnostMystic's user avatar
1 vote
1 answer
216 views

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 ...
dohmatob's user avatar
  • 6,764
4 votes
0 answers
254 views

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(...
dohmatob's user avatar
  • 6,764
4 votes
1 answer
957 views

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 ...
Spencer Kraisler's user avatar
2 votes
1 answer
142 views

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 ...
Uzu Lim's user avatar
  • 841
0 votes
2 answers
181 views

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, ...
dohmatob's user avatar
  • 6,764
2 votes
1 answer
240 views

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'...
ABIM's user avatar
  • 4,775
4 votes
1 answer
207 views

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 +...
dohmatob's user avatar
  • 6,764
2 votes
1 answer
160 views

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 (...
dohmatob's user avatar
  • 6,764
1 vote
0 answers
142 views

$\newcommand\v{\operatorname{vol}_d(C}$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 $\operatorname{vol}_d(C)>0$, ...
dohmatob's user avatar
  • 6,764
0 votes
2 answers
258 views

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 $...
ABIM's user avatar
  • 4,775
1 vote
1 answer
108 views

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 ...
dohmatob's user avatar
  • 6,764
1 vote
0 answers
248 views

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}},...
Satya Prakash's user avatar
1 vote
1 answer
257 views

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(...
Carlos_Petterson's user avatar

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