# Questions tagged [geometric-probability]

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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 $a,b,c$...
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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 $ax + by \ge c$ ...
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### 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 ...
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### 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 ...
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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 $r$ ...
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### 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 ...
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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 ...
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### 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 ...
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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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1 vote
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 ...
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1 vote
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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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1 vote
248 views

### Comparing two multivariate normal distribution

Let $\mathbf{Z}\sim N(\boldsymbol{\mu},\mathrm{\Sigma})$, where \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}},...
1 vote
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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(...