All Questions
Tagged with geometric-probability convex-analysis
9
questions
1
vote
1
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27
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Can we randomly subdivide general convex sets "uniformly" in terms of volume?
I'm not sure how best to phrase my question. I'll explain first what I mean by the title.
If we work on an interval $(a,b)$, then by choosing a number uniformly on this interval we divide the length &...
- 137
1
vote
0
answers
91
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Convergence to an $\ell_p$ ball, of Steiner symmetrization of compact convex subsets of $\mathbb R^n$
Context. I'm working on a problem, and it seems Steiner symmetrization might just be the golden trick. But first, I must make sure the process will converge to an $\ell_p$ ball...
Fix $p \in [1,\...
- 9,575
2
votes
1
answer
534
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Probability that the convex hull of random points is a triangle
Question:
Consider a fixed number $k > 3$ of random points in the plane, each independently distributed according to a 2D standard normal distribution. What is the probability that the convex hull ...
- 19.1k
2
votes
1
answer
1k
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Boyd & Vandenberghe, Exercise 2.15g — Convexity of a region on the probability simplex
Exercise 2.15g of Boyd & Vandenberghe's Convex Optimization:
On the probability simplex in $\mathbb{R}^n$ where each point $p = (p_1, p_2, p_3, \ldots, p_n)$ corresponds to a distribution for ...
- 21
13
votes
1
answer
550
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Expected values of some properties of the convex hull of a random set of points
$N$ points are selected in a uniformly distributed random way in a disk of the unit radius. Let $P(N)$ and $A(N)$ denote the expected perimeter and the expected area of their convex hull.
For what $N$...
- 1,249
1
vote
2
answers
356
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Integral of convex set
Let $D$ be a convex set and $X_1,\dots,X_d$ be integrable random variables. If $X= (X_1,\dots,X_d)$ is in $D$ almost surely,
why is it true that the vector $a= (\mathbb{E}X_1,\dots,\mathbb{E}X_d)$ ...
- 259
4
votes
0
answers
81
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Random convex shapes containing a ball
I'm interested in the properties of randomly generated convex shapes in $n$-dimensional space.
Suppose I were to generate $v$ uniformly distributed random points on the $n$-ball of radius $R$. What ...
- 1,231
21
votes
2
answers
1k
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What is the probability of having a pentagon in 6 points
If the probability that $5$ random points in the plane whose horizontal
coordinate and vertical coordinate are uniformly distributed on the
interval $\left(0,1\right)$ occur to be the vertices of a ...
- 499
18
votes
1
answer
671
views
Expected size of subset forming convex polygon.
If there are $4$ random points in the plane whose horizontal coordinate
and vertical coordinate are uniformly distributed on the interval
$\left(0,1\right)$, what is the expected largest size (or ...
- 499