arXiv subjects

Labrini Hioni

Publications and source records attributed to Labrini Hioni.

3 recordsLinked to original sources

arXiv2016

Asymptotic shape of the convex hull of isotropic log-concave random vectors

Let $x_1,\ldots ,x_N$ be independent random points distributed according to an isotropic log-concave measure $\mu $ on ${\mathbb R}^n$, and consider the random polytope $$K_N:={\rm conv}\{ \pm x_1,\ldots ,\pm x_N\}.$$ We provide sharp estimates for the querma\ss{}integrals and other geometric parameters of $K_N$ in the range $cn\ls N\ls\exp (n)$; these complement previous results from \cite{DGT1} and \cite{DGT} that were given for the range $cn\ls N\ls\exp (\sqrt{n})$. One of the basic new ingredients in our work is a recent result of E.~Milman that determines the mean width of the centroid body $Z_q(\mu )$ of $\mu $ for all $1\ls q\ls n$.

math.MG↗

arXiv2016

Geometry of random sections of isotropic convex bodies

Let $K$ be an isotropic symmetric convex body in ${\mathbb R}^n$. We show that a subspace $F\in G_{n,n-k}$ of codimension $k=\gamma n$, where $\gamma\in (1/\sqrt{n},1)$, satisfies $$K\cap F\subseteq \frac{c}{\gamma }\sqrt{n}L_K (B_2^n\cap F)$$ with probability greater than $1-\exp (-\sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(\mu )$ of an isotropic log-concave probability measure $\mu $ on ${\mathbb R}^n$. For every $1\leq q\leq n$ and $\gamma\in (0,1)$ we show that a random subspace $F\in G_{n,(1-\gamma )n}$ satisfies $Z_q(\mu )\cap F\subseteq c_2(\gamma )\sqrt{q}\,B_2^n\cap F$. We also give bounds on the diameter of random projections of $Z_q(\mu )$ and using them we deduce that if $K$ is an isotropic convex body in ${\mathbb R}^n$ then for a random subspace $F$ of dimension $(\log n)^4$ one has that all directions in $F$ are sub-Gaussian with constant $O(\log^2n)$.

math.MG↗

arXiv2015

Random approximation and the vertex index of convex bodies

We prove that there exists an absolute constant $\alpha >1$ with the following property: if $K$ is a convex body in ${\mathbb R}^n$ whose center of mass is at the origin, then a random subset $X\subset K$ of cardinality ${\rm card}(X)=\lceil\alpha n\rceil $ satisfies with probability greater than $1-e^{-n}$ {K\subseteq c_1n\,{\mathrm conv}(X),} where $c_1>0$ is an absolute constant. As an application we show that the vertex index of any convex body $K$ in ${\mathbb R}^n$ is bounded by $c_2n^2$, where $c_2>0$ is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.

math.MG↗