  In this paper, we introduce several mixed $L_p$ geominimal surface areas for
multiple convex bodies for all $p\neq -n$. 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.
