New Orlicz Affine Isoperimetric Inequalities

Abstract

The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the $L_p$-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz $L_{\phi}$ affine and geominimal surface areas for single convex body as well as for multiple convex bodies, which generalize the $L_p$ (mixed) affine and geominimal surface areas -- fundamental concepts in the $L_p$-Brunn-Minkowski theory. Our extensions are different from the general affine surface areas by Ludwig (in Adv. Math. 224 (2010)). Moreover, our definitions for Orlicz $L_{\phi}$ affine and geominimal surface areas reveal that these affine invariants are essentially the infimum/supremum of $V_{\phi}(K, L^\circ)$, the Orlicz $\phi$-mixed volume of $K$ and the polar body of $L$, where $L$ runs over all star bodies and all convex bodies, respectively, with volume of $L$ equal to the volume of the unit Euclidean ball $B_2^n$. Properties for the Orlicz $L_{\phi}$ affine and geominimal surface areas, such as, affine invariance and monotonicity, are proved. Related Orlicz affine isoperimetric inequalities are also established.