arXiv subjects

Alexandros Eskenazis

Publications and source records attributed to Alexandros Eskenazis.

28 records · Page 2Linked to original sources

arXiv2020

The dimensional Brunn-Minkowski inequality in Gauss space

Let $γ_n$ be the standard Gaussian measure on $\mathbb{R}^n$. We prove that for every symmetric convex sets $K,L$ in $\mathbb{R}^n$ and every $λ\in(0,1)$, $$γ_n(λK+(1-λ)L)^{\frac{1}{n}} \geq λγ_n(K)^{\frac{1}{n}}+(1-λ)γ_n(L)^{\frac{1}{n}},$$ thus settling a problem raised by Gardner and Zvavitch (2010). This is the Gaussian analogue of the classical Brunn-Minkowski inequality for the Lebesgue measure. We also show that, for a fixed $λ\in(0,1)$, equality is attained if and only if $K=L$.

math.MG↗

arXiv2020

Cutoff for the Bernoulli-Laplace urn model with $o(n)$ swaps

We study the mixing time of the $(n,k)$ Bernoulli--Laplace urn model, where $k\in\{0,1,\ldots,n\}$. Consider two urns, each containing $n$ balls, so that when combined they have precisely $n$ red balls and $n$ white balls. At each step of the process choose uniformly at random $k$ balls from the left urn and $k$ balls from the right urn and switch them simultaneously. We show that if $k=o(n)$, this Markov chain exhibits mixing time cutoff at $\frac{n}{4k}\log n$ and window of the order $\frac{n}{k}\log\log n$. This is an extension of a classical theorem of Diaconis and Shahshahani who treated the case $k=1$.

math.PR↗

arXiv2020

On Pisier's inequality for UMD targets

We prove an extension of Pisier's inequality (1986) with a dimension independent constant for vector valued functions whose target spaces satisfy a relaxation of the UMD property.

math.FA↗

arXiv2020

Dimension independent Bernstein-Markov inequalities in Gauss space

We obtain the following dimension independent Bernstein-Markov inequality in Gauss space: for each $1\leq p<\infty$ there exists a constant $C_p>0$ such that for any $k\geq 1$ and all polynomials $P$ on $\mathbb{R}^{k}$ we have $$ \| \nabla P\|_{L^{p}(\mathbb{R}^{k}, \mathrm{d}γ_k)} \leq C_p (\mathrm{deg}\, P)^{\frac{1}{2}+\frac{1}π\arctan\left(\frac{|p-2|}{2\sqrt{p-1}}\right)}\|P\|_{L^{p}(\mathbb{R}^{k}, \mathrm{d}γ_k)}, $$ where $\mathrm{d}γ_k$ is the standard Gaussian measure on $\mathbb{R}^{k}$. We also show that under some mild growth assumptions on any function $B \in C^{2}((0,\infty))\cap C([0,\infty))$ with $B', B''>0$ we have $$ \int_{\mathbb{R}^{k}} B\left( |LP(x)|\right) \mathrm{d}γ_k(x) \leq \int_{\mathbb{R}^{k}} B\left( 10 (\mathrm{deg}P)^{α_{B}}|P(x)|\right)\mathrm{d}γ_k(x) $$ where $L=Δ-x\cdot \nabla $ is the generator of the Ornstein-Uhlenbeck semigroup and $$ α_{B} =1+\frac{2}π \arctan\left(\frac{1}{2}\sqrt{\sup_{s \in (0,\infty)}\left\{\frac{sB''(s)}{B'(s)}+\frac{B'(s)}{sB''(s)}\right\}-2}\right). $$

math.CA↗

arXiv2019

On extremal sections of subspaces of $L_p$

Let $m,n\in\mathbb{N}$ and $p\in(0,\infty)$. For a finite dimensional quasi-normed space $X=(\mathbb{R}^m, \|\cdot\|_X)$, let $$B_p^n(X) = \Big\{ (x_1,\ldots,x_n)\in\big(\mathbb{R}^{m}\big)^n: \ \sum_{i=1}^n \|x_i\|_X^p \leq 1\Big\}.$$ We show that for every $p\in(0,2)$ and $X$ which admits an isometric embedding into $L_p$, the function $$S^{n-1} \ni θ= (θ_1,\ldots,θ_n) \longmapsto \Big| B_p^n(X) \cap\Big\{(x_1,\ldots,x_n)\in \big(\mathbb{R}^{m}\big)^n: \ \sum_{i=1}^n θ_i x_i=0 \Big\} \Big|$$ is a Schur convex function of $(θ_1^2,\ldots,θ_n^2)$, where $|\cdot|$ denotes Lebesgue measure. In particular, it is minimized when $θ=\big(\frac{1}{\sqrt{n}},\ldots,\frac{1}{\sqrt{n}}\big)$ and maximized when $θ=(1,0,\ldots,0)$. This is a consequence of a more general statement about Laplace transforms of norms of suitable Gaussian random vectors which also implies dual estimates for the mean width of projections of the polar body $\big(B_p^n(X)\big)^\circ$ if the unit ball $B_X$ of $X$ is in Lewis' position. Finally, we prove a lower bound for the volume of projections of $B_\infty^n(X)$, where $X=(\mathbb{R}^m,\|\cdot\|_X)$ is an arbitrary quasi-normed space.

math.MG↗

arXiv2019

Gaussian mixtures: entropy and geometric inequalities

A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to $e^{-|t|^p}$ and symmetric $p$-stable random variables, where $p\in(0,2]$. We obtain various sharp moment and entropy comparison estimates for weighted sums of independent Gaussian mixtures and investigate extensions of the B-inequality and the Gaussian correlation inequality in the context of Gaussian mixtures. We also obtain a correlation inequality for symmetric geodesically convex sets in the unit sphere equipped with the normalized surface area measure. We then apply these results to derive sharp constants in Khintchine inequalities for vectors uniformly distributed on the unit balls with respect to $p$-norms and provide short proofs to new and old comparison estimates for geometric parameters of sections and projections of such balls.

math.PR↗

arXiv2019

Nonpositive curvature is not coarsely universal

We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni, Naor and Neiman (2015). We establish this statement by proving that a metric space which is $q$-barycentric for some $q\in [1,\infty)$ has metric cotype $q$ with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the $\ell_\infty$ grid $[m]_\infty^n=(\{1,\ldots,m\}^n,\|\cdot\|_\infty)$ into $\ell_q$ for all $q\in (2,\infty)$, from which we deduce the following discrete converse to the fact that $\ell_\infty^n$ embeds with distortion $O(1)$ into $\ell_q$ for $q=O(\log n)$. A rigidity theorem of Ribe (1976) implies that for every $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ such that if $[m]_\infty^n$ embeds into $\ell_q$ with distortion $O(1)$, then $q$ is necessarily at least a universal constant multiple of $\log n$. Ribe's theorem does not give an explicit upper bound on this $m$, but by the work of Bourgain (1987) it suffices to take $m=n$, and this was the previously best-known estimate for $m$. We show that the above discretization statement actually holds when $m$ is a universal constant.

math.MG↗

arXiv2018

Sharp comparison of moments and the log-concave moment problem

This article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context. In the first part of our work we find the optimal constants in the Khintchine inequality for random vectors uniformly distributed on the unit ball of the space $\ell_q^n$ for $q\in(2,\infty)$, complementing past works that treated $q\in(0,2]\cup\{\infty\}$. As a byproduct of this result, we prove an extremal property for weighted sums of symmetric uniform distributions among all symmetric unimodal distributions. In the second part we provide a one-to-one correspondence between vectors of moments of symmetric log-concave functions and two simple classes of piecewise log-affine functions. These functions are shown to be the unique extremisers of the $p$-th moment functional, under the constraint of a finite number of other moments being fixed, which is a refinement of the description of extremisers provided by the generalised localisation theorem of Fradelizi and Guédon [Adv. Math. 204 (2006) no. 2, 509-529].

math.FA↗

arXiv2013

Topological genericity of nowhere differentiable functions in the disc and polydisc algebras

In this paper we examine functions in the disc algebra $\mathcal{A}(D)$ and the polydisc algebra $\mathcal{A}(D^I)$, where $I$ is a finite or countably infinite set. We prove that, generically, for every $f \in \mathcal{A}(D)$ the continuous periodic functions $u=Ref|_{\mathbb{T}}$ and $\tilde{u} = Imf|_{\mathbb{T}}$ are nowhere differentiable on the unit circle $\mathbb{T}$. Afterwards, we generalize this result by proving that, generically, for every $f \in \mathcal{A}(D^I)$, where $I$ is as above, the continuous periodic functions $u=Ref|_{\mathbb{T}^I}$ and $\tilde{u} = Imf|_{\mathbb{T}^I}$ have no directional derivatives at any point of $\mathbb{T}^I$ and every direction $v \in \mathbb{R}^I$ with $\|v\|_{\infty}=1$.

math.CV↗

arXiv2013

Topological genericity of nowhere differentiable functions in the disc algebra

In this paper we introduce a class of functions contained in the disc algebra $\mathcal{A}(D)$. We study functions $f \in \mathcal{A}(D)$, which have the property that the continuous periodic function $u = Ref|_{\mathbb{T}}$, where $\mathbb{T}$ is the unit circle, is nowhere differentiable. We prove that this class is non-empty and instead, generically, every function $f \in \mathcal{A}(D)$ has the above property. Afterwards, we strengthen this result by proving that, generically, for every function $f \in \mathcal{A}(D)$, both continuous periodic functions $u=Ref|_\mathbb{T}$ and $\tilde{u} = Imf|_\mathbb{T}$ are nowhere differentiable. We avoid any use of the Weierstrass function and we mainly use Baire's Category Theorem.

math.CA↗