IUMJ

Title: On the approximation of a polytope by its dual $L_{p}$-centroid bodies

Authors: Grigoris Paouris and Elisabeth Werner

Issue: Volume 62 (2013), Issue 1, 235-248

Abstract:

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope $P\in\mathbb{R}^n$ by its dual $L_p$-centroid bodies is independent of the geometry of $P$. In particular, we show that if $P$ has volume $1$, \[ \lim_{p\to\infty}\frac{p}{\log p}\left(\frac{|Z_p^{\circ}(P)|}{|P^{\circ}|}-\right)=n^2. \] We provide an application to the approximation of polytopes by uniformly convex sets.