Centroid bodies and comparison of volumes V. YaskinM. Yaskina 52A2046B20convex bodyp-centroid bodyembedding in $L_p$Fourier transform 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 $1$. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2761 10.1512/iumj.2006.55.2761 en Indiana Univ. Math. J. 55 (2006) 1175 - 1194 state-of-the-art mathematics http://iumj.org/access/