In this paper, we prove that an origin-symmetric star body is uniquely determined by its 𝑝-centroid body. Furthermore, using spherical harmonics, we establish a result for non-symmetric star bodies. As an application, we show that there is a unique member of $\Gamma_p\langle K \rangle$ characterized by having larger volume than any other member, for all real $p \geq 1$ that are not even natural numbers, where $\Gamma_p\langle K \rangle$ denotes the 𝑝-centroid equivalence class of the star body 𝐾.

In this paper, the Orlicz mean body $H_{\phi}K$ of a convex body 𝐾 is introduced. Using the notion of shadow system, we establish a sharp lower estimate for the volume ratio of $H_{\phi}K$ and 𝐾.