Inequalities for mixed $p$-affine surface area

Abstract

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed $p$-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of $L_p$ affine surface areas, mixed $p$-affine surface areas and other functionals via these bodies. The surprising new element is that not necessarily convex bodies provide the tool for these interpretations.

39 pages

References in corpus (2)

• New $L_p$ Affine Isoperimetric Inequalities
• A Central Limit Theorem for Convex Sets

Cited by in corpus (6)

• Dual Orlicz-Brunn-Minkowski theory: dual Orlicz $L_{\varphi }$ affine and geominimal surface areas
• New Orlicz Affine Isoperimetric Inequalities
• Chord Measures in Integral Geometry and Their Minkowski Problems
• On $L_p$ Affine Surface Area and Curvature Measures
• $L_p$ geominimal surface areas and their inequalities
• Extremal general affine surface areas