MathSciDoc: An Archive for Mathematician ∫

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.

affine surface area, $L_p$ affine surface area, geominimal surface area, $L_p$ geominimal surface area, $L_p$-Brunn-Minkowski theory, affine isoperimetric inequalities, the Blaschke-Santal\'{o} inequality, the Bourgain-Milman inverse Santal\'o inequality