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

Abstract

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

References in corpus (2)

• Relative entropy of cone measures and $L_p$ centroid bodies
• Mixed $f$-divergence for multiple pairs of measures

Cited by in corpus (3)

• Dual Orlicz-Brunn-Minkowski theory: dual Orlicz $L_{\varphi }$ affine and geominimal surface areas
• New Orlicz Affine Isoperimetric Inequalities
• On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets