The mixed $L_p$ geominimal surface areas for multiple convex bodies

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\mathrm{\ne }-n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are proved. Related inequalities, such as, Alexander-Fenchel type inequality, Santal'{o} style inequality, affine isoperimetric inequalities, and cyclic inequalities are established. Moreover, we also study some properties and inequalities for the $i$-th mixed $L_p$ geominimal surface areas for two convex bodies.