For a star body \(K\) in Euclidean \(n\)-space and for \(-1 < p < 1\) the paper defines polar \(p\)-centroid body \(\Gamma^*_p K\). The authors consider the problem under which assumptions the inclusion \(\Gamma^*_p K \subset \Gamma^*_p L\) implies the inequality \(\text{vol}(L) \leq \text{vol}(K)\). In particular, they show that if \(0<p<1\), then the implication holds true for \(n=2\), but not for \(n \geq 3\). Moreover, if \(-1 < p \leq 0\), then the implication is true if and only if \(n \leq 3\).
The studies of this paper extend some results of E. Lutwak [Proc. Lond. Math. Soc. (3) 60, No. 2, 365–391 (1990; Zbl 0703.52005)] for \(p=1\), and of E. Grinberg and G. Zhang [Proc. Lond. Math. Soc. (3) 78, No. 1, 77–115 (1999; Zbl 0974.52001)] for \(p>1\).