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

TL;DR

This paper investigates how well the volumes of convex symmetric polytopes can be approximated by their dual L_{p}-centroid bodies, revealing a volume convergence rate independent of the polytope's shape.

Contribution

It establishes a universal rate of volume convergence for dual L_{p}-centroid bodies approximating symmetric polytopes, independent of their geometry.

Findings

Volume approximation rate converges as p approaches infinity.

Limit involving p/log p and volume ratios equals n^2.

Application to approximating polytopes with uniformly convex sets.

Abstract

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in 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\rightarrow \infty} \frac{p}{\log{p}} (\frac{|Z_{p}^{\circ}(P)|}{|P^{\circ}|} -1) = n^{2}. We provide an application to the approximation of polytopes by uniformly convex sets.