Lp-sine mixed centroid body and related inequalities

Abstract

In this paper, we define the notion of an $L_p$-sine mixed centroid body ${\Lambda }_{p,i}K$, which is an extension of the $L_p$-sine centroid body ${\Lambda }_{p}K$ introduced by Huang Qingzhong, Li Aijun, Xi Dongmeng, and Ye Deping. Furthermore, we obtain several properties of ${\Lambda }_{p,i}K$, as well as giving two affirmative forms and a negative form of the Shephard type problems, and monotonicity inequalities and Brunn–Minkowski type inequalities for the $L_p$-sine mixed centroid body.