This is an announcement for the paper "Centroids and comparison of volumes" by V.Yaskin and M.Yaskina. Abstract: For $-1<p<1$ 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 $\mathrm{vol}(L)\le \mathrm{vol}(K).$ Our results extend the studies by Lutwak in the case $p=1$ and Grinberg, Zhang in the case $p> 1$. Archive classification: Functional Analysis Mathematics Subject Classification: 52Axx Remarks: 18 pages The source file(s), centr.tex: 51970 bytes, is(are) stored in gzipped form as 0503290.gz with size 13kb. The corresponding postcript file has gzipped size 71kb. Submitted from: yaskinv@math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503290 or http://arXiv.org/abs/math.FA/0503290 or by email in unzipped form by transmitting an empty message with subject line uget 0503290 or in gzipped form by using subject line get 0503290 to: math@arXiv.org.