The (φ,ψ) Orlicz mixed Petty bodies

Abstract

In this paper, the homogeneous (φ,ψ) Orlicz mixed affine and geominimal surface areas are introduced. Their homogeneity, affine invariance and affine isoperimetric inequalities are established. Under some assumptions, the existence and uniqueness of the homogeneous (φ,ψ) Orlicz mixed Petty bodies are established. Base on these, the continuity of the homogeneous Orlicz mixed geominimal surface areas is given. Moreover, the nonhomogeneous (φ,ψ) Orlicz mixed affine and geominimal surface areas are defined, and the continuity of the nonhomogeneous (φ,ψ) Orlicz mixed geominimal surface for some special φ and ψ is obtained as well.