All Questions
Tagged with geometric-measure-theory geometric-probability
15 questions
3
votes
1
answer
181
views
Probability measure on partition theorem
Probability measure in $\mathbb{R}^1$: Continuous function $f$ that is positive everywhere $\ \int_{-\infty}^{\infty} f dx =1$ (total area under the curve equals 1). For visualization see here.
...
- 53
1
vote
0
answers
98
views
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 ...
- 633
1
vote
1
answer
115
views
Approximating the probability of a half-space using random Voronoi diagrams
Fix a half-space $H = \{x_1 \geq 0: ~ (x_1,\dots,x_n) \in \mathbb{R}^n\}$. Let $p$ be a distribution with support in $\mathbb{R}^n$. I am interested in the following way of estimating the weight $p(H) ...
- 33
2
votes
1
answer
297
views
Examples of "almost" Ahlfors regular measures
Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ such that there are $c,C,d,D>0$ satisfying: for every $x \in \mathbb{R}^n$ and every $r>0$
$$
c r^d \leq \mu(B(x,r)) \leq Cr^D.
$$
Let'...
- 5,405
1
vote
1
answer
306
views
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(...
- 195
0
votes
0
answers
131
views
Barycenters on Hadamard Manifolds
Let $(M,g,m_0)$ be a pointed-Hadamard manifold with Riemmanian distance function $d_g$, $(X,\Sigma,\mu)$ be a finite measure space. We use $L^2(\mu;M,m_0)$ to denote the metric space consisting of ...
0
votes
1
answer
132
views
Projection onto manifold of Gaussian measures by "trunction" of moments
Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^n$ with finite mean and variance; in the sense that
$$
\int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \...
- 5,405
2
votes
0
answers
248
views
Intrinsic volume - is there a simplified formula?
I'm struggling to understand how can I compute the instrinsic volumes (or Minkowski functionals) of a submanifold $\Omega$ of $\mathbb{R}^N$. I found a formula, but I really can't understand it...
It ...
- 1,759
0
votes
0
answers
156
views
Function classes with high Rademacher complexity
My question is two fold,
Is there any general understanding of what makes a function class have high Rademacher complexity? (Sudakov minoration would say that one sufficient condition for a class of ...
- 2,246
1
vote
0
answers
259
views
Is there a precise relationship between ``Geometric Functional Analysis" and high-dimensional probability/information theory?
The 2009 course on GFA by Roman Vershynin (https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf) introduced the subject with this line on the course page, "...
- 2,246
1
vote
0
answers
94
views
Measure of the boundary of the support of a certain function defined by an expectation
Suppose:
$\mathcal{S} = \{ S \in \mathbb{R}^d \ | \ S_i > 0, \forall i = 1,...,d \} $
$R$ is a random vector (on some probability space, $\Omega$) such that, $R: \Omega \to \mathcal{S}$.
$h : ...
- 111
3
votes
2
answers
355
views
Concentration of norm of projection onto a subspace
Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. Let $V$ be a linear subspace of dimension $k$ and let $P_V(x)$ be the orthogonal projection of $x$ onto $V$.
I ...
- 398
2
votes
0
answers
899
views
norm of projection of a random vector on the sphere onto a linear subspace
Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{D-1}$ and let $\mathcal{V}$ be a linear subspace of dimension $m$. Then it is known that the euclidean norm of the ...
- 398
1
vote
1
answer
150
views
Probability Content of a random ball in R^n
As a follow up to this question, concerning this paper:
Given random variables $X_1,\ldots,X_N,X_q:\Omega\rightarrow\mathbb{R}^d$, where $X_1,\ldots,X_N$ are independent and identical distributed. ...
- 329
2
votes
2
answers
425
views
understanding the average height of a unit hyper-semisphere
Consider the quantity
$$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$$
where $S_{n-2}$, $V_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb R^{n-1}$.
...
- 398