The general minimal Orlicz surface area for measures

Abstract

Petty proved that a convex body in [Formula: see text] has the minimal surface area among its [Formula: see text] images if and only if its surface area measure is isotropic. In this paper, we seek the general minimal Orlicz surface area of a convex body containing the origin in its interior among its [Formula: see text] images for a measure [Formula: see text] on [Formula: see text] with density that has a positive degree of homogeneous, demonstrate the existence of the extreme body, and establish a necessary condition for the extreme body. Moreover, we also provide a counterexample to negate the uniqueness of the extreme body attaining the general minimal Orlicz surface area for [Formula: see text].