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Volume estimates for some random convex sets

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Convex and discrete geometry; Functional analysis

video/mp4

Author affiliation: National and Kapodistrian University of Athens

2020-08-10

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/75400

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/