All Questions
Tagged with convex-geometry pr.probability
69 questions
5
votes
3
answers
7k
views
Estimate probability( 0 is in the convex hull of N random points ) ?
Can anyone estimate N such that Prob( 0 is in the convex hull of $N$ points ) >= .95
for points uniformly scatterered in $[-1,1]^d$, $d = 2, 3, 4, 10$ ?
The application is nearest-neghbour ...
- 265
1
vote
1
answer
234
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 $...
- 773
4
votes
0
answers
93
views
On symmetry and measure concentration rate for convex bodies
The concentration of measure on the cube $ [0, 1]^n $ equipped with uniform probability measure $\mu_{\infty}$,
states that for any $A \subset [0, 1]^n $ with $ \mu_{\infty}(A) \geq \frac{1}{2} $,
we ...
- 365
3
votes
1
answer
188
views
Equivalent Definitions of the Gaussian Surface Measure for Regular Sets
I wonder if the following definitions of the Gaussian surface measure are equivalent.
First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....
- 1,127
6
votes
2
answers
413
views
Random walk in a convex body or convex polytope
Let $\Delta$ be a convex body (i.e. a compact convex subset) or a convex polytope in
$\mathbb{R}^n$. Let $x$ be a point inside $\Delta$ and consider a (uniform) random walk starting at $x$ inside $\...
- 893
3
votes
1
answer
213
views
A lottery on coins in a convex set
You play the following game.
You get $4n$ gold coins and have to arrange them in the unit square in general position (no two coins have the same x or the same y coordinate). Call this set of coins $...
- 3,549
4
votes
1
answer
227
views
Matching moments in even dimensions
Let $D$ be a probability distribution on the unit interval $[0,1]$ with moments $\mu_i=\mathbb{E}_D [x^i]$. Let $\delta(x)$ be a singleton probability distribution with all weight at $x\in [0,1]$. ...
- 3,979
7
votes
1
answer
444
views
Is an infinite-dimensional "Lebesgue measure" uniquely determined by a set of positive finite measure?
Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex.
We say that $\mu$ admits shifts if ...
- 4,010
5
votes
2
answers
985
views
Expected distance of a random point to the convex hull of N other points
Let $X_1, \cdots, X_N$ be i.i.d. $d$-dimensional random vectors, the exact distribution of $X$ is not very important in my application (as long as it continuous) so pick the one that works best, but ...
- 51
1
vote
0
answers
34
views
Limiting law of quadratic functions of sample averages
Let $X_1,\cdots,X_n$ be independent centered univariate random variables. Let also $\{w_{ij}\}_{i,j=1}^{k,n}$ be a set of deterministic scalar weights, where $k\ll n$. Define sample averages
$$
\...
- 373
3
votes
0
answers
134
views
Algorithm to calculate moments of uniform distribution on convex polyhedra
There is system of linear inequalities
$$
Ax \leq K,
$$
$$
x\geq a, x\leq b.
$$
$A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$.
Suppose that on set of solutions ...
7
votes
3
answers
377
views
Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$
Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly ...
- 151k
2
votes
0
answers
79
views
Compute Mixed Volume with Respect to Some Regular Sets
Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
- 1,127
3
votes
0
answers
635
views
Wasserstein distance, convex polytopes and extreme points
Let us consider convex polytopes with $K$ extreme points in $\mathbb{R}^d$. Let $\mathbf{P}$ be such polytope, and let $\text{ext}(\mathbf{P})$ denote its set of extreme points, so that $\mathbf{P} = \...
- 1,731
2
votes
3
answers
1k
views
Draw a Random Line Through a Voronoi Tessellation, What is the Average Number of Voronoi Cell the Line Intersects?
Update: problem reformulation
Following the advice in comments, I now restate my problem using Voronoi
tessellation.
Given a unit hypercube $H_n=\{(x_1,\ldots,x_n)\in \mathbb{R}^n: 0\leq x_i\leq
1\}$...
4
votes
1
answer
503
views
An elementary probability question
Let $X$ be a $d$-dimensional random vector distributed according to probability measure $D$. At least the second moment of the coordinates of $X$ is finite.
Consider $n+1$ samples $X_0, \ldots, X_n ...
- 461
1
vote
0
answers
93
views
Bounds on the spherical measure of sub-level sets of quadratic forms
I'm wondering if there are any bounds on the spherical measure of sets of the form
$$
\mu_n\left(\{y\in S^{n-1} : \frac{y_1^2}{y_2^2} < \alpha\}\right) \leq f(\alpha)
$$
where $\alpha$ is some ...
- 205
2
votes
0
answers
399
views
Convexity and probability
Problem instance: A closed convex body $B\subset {\Bbb R}^n$ of volume 1; a point $p\in B$; and a real number $v\in(0,1)$.
Objective: Find the probability $P(B,v,p)$ that $p\in B'$, for $B'$ a ...
- 17.6k
13
votes
0
answers
1k
views
Constructive aspects of Caratheodory's theorem in convex analysis
Let me paraphrase Caratheodory's theorem in a probabilistic setup:
Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
- 1,839