  In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq
p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and
the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$
geominimal surface area is motivated by recent work on the extension of the
$L_p$ affine surface area -- a fundamental object in (affine) convex geometry.
We prove some properties for the $L_p$ geominimal surface area and its related
inequalities, such as, the affine isoperimetric inequality and a Santal\'{o}
style inequality. Cyclic inequalities are established to obtain the
monotonicity of the $L_p$ geominimal surface areas. Comparison between the
$L_p$ geominimal surface area and the $p$-surface area is also provided.
