The Minimal Dual Orlicz Surface Area

Abstract

Petty proved that a convex body in $\mathbb{R}^{n}$ has the minimal surface area among its $\operatorname{SL}(n)$ images if and only if its surface area measure is isotropic. Recently, Zou and Xiong generalized this result to the Orlicz setting by introducing a new notion of minimal Orlicz surface area, and the analog of Ball's reverse isoperimetric inequality is established. In this paper, we give the dual results in Orlicz space by introducing a new notion of minimal dual Orlicz surface area. And the dual form of Ball's isoperimetric inequality is established.