New Orlicz affine isoperimetric inequalities

The Orlicz–Brunn–Minkowski theory received considerable attention recently, and many results in the LpLp-Brunn–Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz LϕLϕ affine and geominimal surface areas for a single convex body as well as for multiple convex bodies, which generalize the LpLp (mixed) affine and geominimal surface areas – fundamental concepts in the LpLp-Brunn–Minkowski theory. Our extensions are different from the general affine surface areas by Ludwig in [21]. Moreover, our definitions for Orlicz LϕLϕ affine and geominimal surface areas reveal that these affine invariants are essentially the infimum/supremum of Vϕ(K,L∘)Vϕ(K,L∘), the Orlicz ϕ-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 B2n. Properties for the Orlicz LϕLϕ affine and geominimal surface areas, such as affine invariance and monotonicity, are proved. Related Orlicz affine isoperimetric inequalities are also established.