Lp Geominimal Surface Areas and their Inequalities

In this paper, we introduce the L
p
geominimal surface area for all −n≠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 Santaló 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.