Title: Centroid bodies and comparison of volumes

Authors: V. Yaskin and M. Yaskina

Issue: Volume 55 (2006), Issue 3, 1175-1194

Abstract: For $-1$ less than $p$ less than $1$ we introduce the concept of a polar $p$-centroid body $\Gamma^{*}_p K$ of a star body $K$. We consider the question of whether $\Gamma^{*}_p K \subset \Gamma^{*}_p L$ implies $\mbox{vol}(L) \le \mbox{vol}(K)$. Our results extend the studies by Lutwak in the case $p = 1$ and Grinberg, Zhang in the case $p$ greater than $1$.