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A question on Ibragimov's theorem on strong unimodality

I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
Ervand's user avatar
  • 51
1 vote
0 answers
143 views

Integer points inside the high-dimensional ball (asymptotics)

Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely: $$...
DJA's user avatar
  • 435
1 vote
0 answers
45 views

Inequality Involving Concave Monotonic Function

Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...
Alireza Bakhtiari's user avatar
0 votes
0 answers
72 views

Probability of being inside a convex n-dimensional polytop

I am currently conducting some post-grad research about wireless transmissions with uncertain transmission delays. As part of the research, each individual transmission is modelled using a probability ...
Florian Bauer's user avatar
1 vote
0 answers
50 views

Interpolation in convex hull

I'm reading a paper, Learning in High Dimension Always Amounts to Extrapolation, that provides a result I don't understand. It provides this theorem which I do understand: Theorem 1: (Bárány and ...
Christopher D'Arcy's user avatar
0 votes
0 answers
22 views

Directions of differentiability of log-concave measures with infinite-dimensional support

I recently came across the very nice review "Differentiable Measures and the Malliavin Calculus" by Bogachev (1997) which begins by discussing measures $\mu$ on locally convex spaces $X$ ...
iolo's user avatar
  • 651
0 votes
0 answers
43 views

For a convex body $K\subset R^n$, does the quantity $\min_{t>0} E[t^{-1} \|tg - \pi_K( t g)\|_2]$ have a name? Where has it been studied?

Consider a convex compact set $K\subset R^n$ (with non-empty interior if that helps). Let $\Pi_K:R^n\to K$ be the projection onto $K$, defined as $$ \Pi_K(x) = \operatorname{argmin}_{k\in K} \|k-x\| $$...
jlewk's user avatar
  • 1,724
1 vote
0 answers
245 views

A sort of dual to nondegenerate random variables

I was motivated by this classical puzzle/1992 Putnam problem. Suppose 4 points are independently and uniformly distributed on a sphere in 3d. What is the probability the tetrahedron they form contains ...
Jess Boling's user avatar
1 vote
1 answer
195 views

Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand

I'm looking for references for two facts that are stated without proof in the paper: Talagrand, M., Are all sets of positive measure essentially convex?, Lindenstrauss, J. (ed.) et al., Geometric ...
Samuel Johnston's user avatar
2 votes
0 answers
164 views

Log Sobolev inequality for log concave perturbations of uniform measure

Suppose $\Omega$ is a convex bounded open set of $\mathbb{R}^n$ (I would be happy with just $\Omega$ as the $n$-dimensional cube). Let $\mu$ be the uniform measure on $\Omega$ and consider the ...
Matt Rosenzweig's user avatar
1 vote
0 answers
61 views

$\psi_2$ marginals of the permutahedron?

Let $K$ be a convex body. I in particular care about the permutahedron. I will view this as being the convex hull of all coordinate-wise permutations of the vector $$v = \frac{1}{2n+2}(-n, -n+2,\dots, ...
Mark Schultz-Wu's user avatar
5 votes
0 answers
158 views

Log Sobolev inequality uniform in parameters

Fix a positive integer $N$. For $\theta \in [0,2\pi]$, set $\sigma_k(\theta) :=(\cos(k\theta),\sin(k\theta)) \in S^1$ for each integer $1\leq k\leq N$. Now for vectors $x_1,\ldots,x_N\in \mathbb{R}^2$,...
Matt Rosenzweig's user avatar
1 vote
0 answers
83 views

Closed form volumes for intersecting modified cylinders

This question is somewhat related to the question Intersecting cylinders, but where the cylinders are now modified to orbifolds in the hypercube with singularities occurring at the vertices of the ...
John McManus's user avatar
1 vote
0 answers
81 views

Expectation of dual norm induced by probability measure

Let $\mu$ be a probability measure supported on the unit sphere $\mathbb{S}^{d-1}$. Assuming that $\mu$ is even and not supported on any great subsphere, its cosine transform $w \in \mathbb{R}^d \...
sbnietert's user avatar
  • 103
1 vote
1 answer
176 views

Is an inner product $\langle X, \epsilon\rangle$ between log-concave $X$ and $\epsilon\gets \{0,1\}^n$ log concave?

Let $X$ be a random variable with a density $p(x)$ with respect to the Lebesgue measure. We say that $X$ is log concave if $p(x) = \exp(-V(x))dx$ for $V(x)$ a convex function. Let $X$ be log-concave ...
Mark Schultz-Wu's user avatar
1 vote
1 answer
176 views

Tight upper-bounds for the Gaussian width of intersection of intersection of hyper-ellipsoid and unit-ball

Let $\Lambda$ be a positive-definite matrix of size $n$ and let $R \ge 0$, which may depend on $n$. Consider the set $S := \{x \in \mathbb R^n \mid \|x\|_2 \le R,\,\|x\|_{\Lambda^{-1}} \le 1\}$ where $...
dohmatob's user avatar
  • 6,853
0 votes
0 answers
140 views

Anti-concentration of the $\ell_2$ norm of log-concave measures

This question is regarding a special case of this question, for which it is plausible the details are known. The Carbery-Wright inequality is an "anti-concentration inequality" that states ...
Mark Schultz-Wu's user avatar
1 vote
1 answer
231 views

For $x_1,...,x_n$ iid random on sphere of radius $\sqrt{d}$ in $R^d$, what is a good upper-bound on min distance of $x_{n}$ from the other $x_i$'s?

Let $n$ and $d$ be large positive integers with $n \le d^\gamma$, for some absolute constant $\gamma>0$; i.e., $n$ is at most polynomial in $d$. Let $x_1,\ldots,x_n,x_{n+1}$ be drawn iid from the ...
dohmatob's user avatar
  • 6,853
0 votes
0 answers
118 views

Weak derivative of projection onto probabilist's simplex

Let $\Delta_n:=\{x\in [0,1]^n:\boldsymbol{1}^{\top}x=1\}$ denote the probabilist's $n$-simplex and let $P:\mathbb{R}^n\rightarrow\Delta_n$ denote the (Euclidean) metric projection onto this simplex ...
ABIM's user avatar
  • 5,405
2 votes
1 answer
165 views

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 (...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
113 views

Existence of fine approximate of a convex body in $\mathbb R^d$ with convex hull of $\mathcal O(d)$ points

Let $K$ be a convex body in $\mathbb R^d$ which contains the origin and let $\theta \in (0,1)$. Question. Is it always possible to find $n$ points $x_1,\dotsc,x_n \in \mathbb R^d$ such that $$ \theta ...
dohmatob's user avatar
  • 6,853
7 votes
4 answers
476 views

What does $\mathbb E_V \max_{x \in V,\,\|x\|=1} x^T Ax$ evaluate to when $V$ is random $k$-dim suspace of $\mathbb R^n$ and $A$ is fixed psd matrix?

Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant–Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by $...
dohmatob's user avatar
  • 6,853
1 vote
0 answers
153 views

Is there a polynomial expression for the volume of the following set?

Denote the unit $\ell_2$ ball in $\mathbb{R}^n$ as $\mathcal{B}_n$. It is widely kown that for a convex body $\mathcal{K}\subseteq \mathbb{R}^n$, the $n$-dimensional volume of the parallel body $\...
RyanChan's user avatar
  • 550
1 vote
1 answer
67 views

Lower-bound for $\mathbb E[e^{-b(v^\top X - c)^2}]$, when $X$ is log-concave in high-dimensions

Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that ...
dohmatob's user avatar
  • 6,853
6 votes
3 answers
447 views

Isoperimetric inequality for $\epsilon$-expansion of a set only along a certain subspace

Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = ...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
133 views

Discrete random walk on polytope via involutions

Let $P$ be a convex polytope (or more generally convex body, I suppose) in $V=\mathbb{R}^n$. For each $v\in \mathbb{P}V$, we define an involution $\tau_v\colon P\to P$ by setting $\tau_v(p)$ to be the ...
Sam Hopkins's user avatar
  • 24.2k
4 votes
0 answers
116 views

Improving log-Sobolev inequalities via quadratic regularisation

Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$. For suitable functions $g \geqslant 0$, define $$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...
πr8's user avatar
  • 801
0 votes
2 answers
532 views

Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$

Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$. Question. What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
114 views

Reweighting probability measures by convex potentials, and contraction in transport distance

Let $W: \mathbf{R}^d \to \mathbf{R}$ be a convex function such that $\int \exp(-W) = 1$, and define probability measures $\mu_y$ by $$\mu_y (dx) = \exp( - W (x - y)) \,dx,$$ i.e. each $\mu_y$ is a ...
πr8's user avatar
  • 801
1 vote
0 answers
45 views

Analytic lower-bound for minimal value of $\|x\|^2$ such that $\|Cx-b\|^2 \le c^2$ (a hyperellipsoid)

Let $C$ be an $n \times p$ matrix and $b$ be a column vector of length $n$, and $c>0$. Let $E := \{x \in \mathbb R^p \mid \|Cx-b\| \le c\}$, a hyperellipsoid in nonstandard position. Question 1. ...
dohmatob's user avatar
  • 6,853
3 votes
0 answers
83 views

Ornstein-Ulhenbeck like semigroup for the orthogonal group with a hypercontractive inequality

I am looking for an Ornstein-Uhlenbeck like semigroup $P_t$ and associated generator $\mathcal{L}$ on $G = \operatorname{SO}(n)$ or $\operatorname{O}(n)$ that has a hypercontractive inequality with a ...
arjun's user avatar
  • 941
2 votes
0 answers
106 views

How to judge whether the following convex set contains a given point?

Let the set $\mathcal{S}=\left\{ \sum_{i=1}^n x_i\mathbf{h}_i:x_i\in[0,1] \text{ for all }i\right\}\subset\mathbb{R}^r$, i.e., a zonotope generated by $n$ column vector $\mathbf{h_1},\cdots,\mathbf{h}...
RyanChan's user avatar
  • 550
2 votes
2 answers
308 views

Expected value of Tukey’s half-space depth for log-concave measures

Let ${\mathbb P}$ be a probability measure in ${\mathbb R}^n$. Let $x\in{\mathbb R}^n$ be an arbitrary point. Let ${\mathbb H}_x$ be the set of halfspaces of ${\mathbb R}^n$ containing $x$. Let \begin{...
Bogdan's user avatar
  • 781
2 votes
0 answers
104 views

Weak convergence rates for integral operators

Suppose $q=\sum_{i=1}^m\pi_i\delta_{x_i}$ is a discrete measure on $\mathbb{R}^n$ and let $q\ast \varphi_\epsilon$ denote the convolution of $q$ with some mollifier $\varphi_\epsilon$, so that $q\ast\...
Jeff S's user avatar
  • 75
1 vote
1 answer
59 views

Characterization of random variables whose tensor powers have subexponential "small-ball" probabilities

Is there a succinct characterization of all random variables $\zeta$ on $\mathbb R$ with the following properties 1. Symmetry: $\zeta \overset{d}{=} - \zeta$. 2. Small-ball probability: there exists ...
dohmatob's user avatar
  • 6,853
1 vote
0 answers
83 views

Given a large random matrix, how to prove that every large submatrix whose range contains a large ball?

Context. Studing a problem in machine-learning, I'm led to consider the following problem in RMT... Definition. Given positive integers $m$ and $n$ and positive real numbers $c_1$ and $c_2$, let's ...
dohmatob's user avatar
  • 6,853
4 votes
0 answers
116 views

Log-Sobolev Inequalities for convex bodies

For a measure $\mu$ supported on a convex body $K$, what are the conditions on $\mu$ and $K$ to satisfy a Log-Sobolev inequality of the form: $$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\...
Kcafe's user avatar
  • 519
1 vote
0 answers
248 views

Gaussian mean width of normal random cones

Suppose $1 \leq n < m < \infty$ are integers. For $g \sim \mathcal N(0, I_n)$ define the gaussian mean width of a non-empty set $T \subseteq \mathbb R^n$ by $$ w(T) := \mathbb E \sup_{x \in T} \...
bashfuloctopus's user avatar
8 votes
1 answer
355 views

Lower Bound of KL-Divergence Between Two Gibbs Measures

Suppose we have two Gibbs measures with densities $$ p_f(x) \propto \exp(f(x)),\quad q_g(x)\propto \exp(g(x)). $$ Consider the KL-divergence between $p_f$ and $q_g$, as a functional of $f$ and $g$, ...
Minkov's user avatar
  • 1,127
4 votes
0 answers
156 views

Geometric meaning of the chi-square "measure of association"

In Statistics, there's a standard test of independence of two random variables taking values in finite sets $I,J$. It relies on the computation of $\chi$-square statistics, $$ \chi^2:=\sum_{(i,j)\in ...
Kostya_I's user avatar
  • 8,992
0 votes
0 answers
165 views

Probability that the perturbed convex hull is larger than the original one

I am wondering if any convex geometers/probabilists have looked at the following question: Given $n$ randomly distributed (not sure what assumption to put there) points in $\mathbb{R}^d$, for each ...
user3799934's user avatar
3 votes
3 answers
439 views

Inner radius of a random convex hull

Let $\sigma_1,\ldots,\sigma_M$ i.i.d. random vectors in $\mathbb{R}^d$, and for notational convenience, let $\Sigma=(\sigma_1,\ldots,\sigma_M)$. I am interested in understanding $$ \gamma(\Sigma) = \...
Cristóbal Guzmán's user avatar
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 $$ \...
Yining Wang's user avatar
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 ...
random_shape's user avatar
4 votes
1 answer
290 views

On the 1/2 assumption on concentration of measure for continuous cube

The concentration of measure on $ [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 have: $$...
random_shape's user avatar
9 votes
2 answers
987 views

Average size of extreme points of convex hull of $N$ points

Fix $n$ a (small) integer. Let $N$ be a (big) integer. Consider $N$ random points in the $n$-dimensional unit cube $[0, 1]^n$. The $N$ points are independently uniformly distributed. Define $V(N)$ ...
WhatsUp's user avatar
  • 3,432
14 votes
1 answer
449 views

References for reasoning about the spectrum of a convex body?

By "spectrum of a convex body", I mean: start with a convex body $B$ in $\mathbb{R}^d$, then consider the corresponding $d \times d$ covariance matrix resulting from a uniform distribution over $B$ -- ...
Barbot's user avatar
  • 143
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 $...
John Wong's user avatar
  • 773
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 ...
Steve's user avatar
  • 1,127
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....
Steve's user avatar
  • 1,127