All Questions
Tagged with geometric-probability convex-geometry
14
questions
2
votes
1
answer
256
views
Does the Mean of a Convex Body Shift when the Body Shifts?
Suppose you put a strictly positive (supported on all $\mathbb{R}^n$) probability measure $\psi$ on $\mathbb{R}^n$. Suppose its density has only one local maximum from which the density decreases in ...
- 630
1
vote
0
answers
109
views
Projection of a Gaussian random vector onto the unit $\ell_1$ ball
Let $Z_n \in \mathbb{R}^n$, $Z_n \sim N(0, I_n)$ be a gaussian random vector, where $I_n$ is the identity matrix. The unit ball is defined as
$$ L_1 = \left[X \in \mathbb{R}^n: \| X \|_1 \leq 1 \...
2
votes
1
answer
60
views
Magnitude Of Spherical Simplex Centroid Is Decreasing
Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...
- 630
1
vote
1
answer
27
views
Can we randomly subdivide general convex sets "uniformly" in terms of volume?
I'm not sure how best to phrase my question. I'll explain first what I mean by the title.
If we work on an interval $(a,b)$, then by choosing a number uniformly on this interval we divide the length &...
- 137
1
vote
1
answer
111
views
Expected number of uniformly random points in unit square is in convex position.
If $n$ points are uniformly generated in a unit square, following famous Erdős–Szekeres theorem.
The probability of them be in convex position is ${\displaystyle \left({\binom {2n-2}{n-1}}/n!\right)^{...
- 1,271
1
vote
0
answers
48
views
Concentration: lower bound for $\sup_u \mu(\{a + \epsilon u \mid a \in A\})$, where the sup is over unit vectors $u \in \mathbb R^n$
Let $\mu$ be a probability distribution on $\mathbb R^n$. For $\epsilon > 0$ and a Borell set $A \subseteq R^n$ with $\mu(A) > 0$, define the $\epsilon$-neighborhood of $A$ as $A^\epsilon := \{...
- 9,575
1
vote
0
answers
91
views
Convergence to an $\ell_p$ ball, of Steiner symmetrization of compact convex subsets of $\mathbb R^n$
Context. I'm working on a problem, and it seems Steiner symmetrization might just be the golden trick. But first, I must make sure the process will converge to an $\ell_p$ ball...
Fix $p \in [1,\...
- 9,575
2
votes
0
answers
195
views
When is the measure of spherical cap large?
It is known that in high dimensions, the measure of the spherical cap is small, due to the measure concentration for the sphere. In particular, we have the following inequality in $n$ dimension:
$$
1-\...
- 159
9
votes
1
answer
258
views
Continuity of the Euler characteristic with respect to the Hausdorff metric
Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means ...
1
vote
0
answers
32
views
Valuation Property for mean width
For some polyhedron, $P$, define the mean width function,
$$H(P)=\sum_{e\in E} L_e(\pi - \delta_e)/(4\pi)$$
Where $E$ is the set of all edges of the polyhedron, $L_e$ is the length of edge $e$ and $\...
- 766
4
votes
1
answer
524
views
Estimate the volume of Voronoi cell
Let given a ball of radius $\alpha$ centered in point $u$ in $d$-dimensional space. Let given a sample of $n$ uniformly distributed vectors $x_i$ ($i = 1,\dots,n$) inside the ball. For each vector $...
- 1,024
0
votes
0
answers
33
views
Lower-bounding Gaussian inner products with high probability
Suppose that $K\subseteq \mathbb R^n$ is a proper convex set with piecewise smooth boundary and that $0 \in K$. Assume that $x \in K$ and let $z \sim \mathcal{N}(0, I_n)$ be a Gaussian random vector. ...
- 673
1
vote
1
answer
158
views
Defining the "Level of Convexity" for Non-Convex Bodies
The definition of convexity for a body in $\mathbb{R}^n$ is simple enough, namely: a body $K\subset \mathbb{R}^n$ is convex if for any two points $x,y\in K$ the line segment between $x$ and $y$ is ...
- 58
0
votes
0
answers
304
views
What is a random set of points in $R^2$?
Given a finite set of $n$ points $S$ in $R^2$, its convex hull, $cvx(S)$, can be obtained with the aid of many algorithms. To numerically compare these algorithms and study their complexity I need to ...
- 3,702