# Open Collections

## BIRS Workshop Lecture Videos ## BIRS Workshop Lecture Videos

### Volume estimates for some random convex sets Giannopoulos, Apostolos

#### Description

Our aim is to provide some estimates on the expected volume of various random convex sets, combining rearrangement inequalities and other tools from asymptotic geometric analysis:  $\bullet$ For any ${\bf x}=(x_1,\ldots ,x_N)\in \oplus_{i=1}^N{\mathbb R}^n$ we denote by $T_{{\bf x}}=[x_1\cdots x_N]$ the $n\times N$ matrix with columns $x_1,\ldots ,x_N$. We discuss upper and lower bounds for the expected volume $${\mathbb E}_{\mu^N}\big({\rm vol}_n(T_{{\bf x}}(K))\big):= \int_{{\mathbb R}^n}\cdots \int_{{\mathbb R}^n}\big({\rm vol}_n(T_{{\bf x}}(K))\big)\,d\mu (x_N)\cdots d\mu (x_1)$$ of $T_{{\bf x}}(K)$, where $K$ is a centrally symmetric convex body in ${\mathbb R}^N$ and $\mu$ is an isotropic log-concave probability measure on ${\mathbb R}^n$.  $\bullet$ Let $K$ be a centrally symmetric convex body in ${\mathbb R}^n$ and let $x_1,\ldots ,x_N$ be independent random points uniformly distributed in $K$. Given $r_1,\ldots ,r_N>0$, we discuss upper and lower bounds for the expected volume of the random polyhedron $\bigcap_{i=1}^NB(x_i,r_i)$.  If time permits, we shall also discuss a question of V.~Milman asking for the equivalence of the norms \begin{equation*}\|{\bf t}\|_{K^s,K}=\int_{K}\cdots\int_{K}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_s\cdots dx_1\approx \|{\bf t}\|_2\end{equation*} for all ${\bf t}\in {\mathbb R}^s$, where $K$ is a centrally symmetric convex body of volume $1$ in ${\mathbb R}^n$.  The talk is based on joint works with G. Chasapis and N. Skarmogiannis.