The mixed $L_p$ geominimal surface areas for multiple convex bodies Deping YeJiazu ZhouBaocheng Zhu 52A2053A15Mixed volumeAlexander-Fenchel inequalityMinkowski inequalityaffine isoperimetric inequalitiesthe Blaschke-Santal\\\'{o} inequalitythe Bourgain-Milman inverse Santal\\\'o inequality$L_p$ affine surface area$L_p$ geominimal surface areamixe In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $-n\neq p\in\mathbb{R}$. Our definitions are motivated by an equivalent formula for the mixed $p$-affine surface area. Properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are proved. Related Alexander-Fenchel type inequality, Santal\'o style inequalities, affine isoperimetric inequalities, and cyclic inequalities are established. Moreover, we also study properties and inequalities for the $i$-th mixed $L_p$ geominimal surface areas for two convex bodies. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5623 10.1512/iumj.2015.64.5623 en Indiana Univ. Math. J. 64 (2015) 1513 - 1552 state-of-the-art mathematics http://iumj.org/access/