Centroids and comparison of volumes

Abstract

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

18 pages