Associated with the $L_p$-curvature image defined by Lutwak, some inequalities for extended mixed 𝑝-affine surface areas of convex bodies and the support functions of $L_p$-projection bodies are established. As a natural extension of a result due to Lutwak, an $L_p$-type affine isoperimetric inequality, whose special cases are $L_p$-Busemann–Petty centroid inequality and $L_p$-affine projection inequality, respectively, is established. Some $L_p$-mixed volume inequalities involving $L_p$-projection bodies are also established.

We obtain an isoperimetric inequality which estimate the affine invariant 𝑝-surface area measure on convex bodies. We also establish the reverse version of $L_p$-Petty projection inequality and an affine isoperimetric inequality of $\Gamma_{-p}K$.

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.