# Questions tagged [geometric-probability]

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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### 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, $f_1,f_2 \in F$, ...
1 vote
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### 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 ...
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### 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 (...
1 vote
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### $\newcommand\v{\operatorname{vol}_d(C}$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 $\operatorname{vol}_d(C)>0$, ...
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### Average distance of the mean of $n$ random complex numbers in a unit disc

Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers distributed uniformly and randomly over the unit disc $x^2+y^2 \leq 1$. Let $z$ be the complex number defined by the mean of the of these numbers,that ...
### Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere
Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...