(p,q)-Mixed geominimal surface area and (p,q)-mixed affine surface area

Abstract In this paper, the ( p , q ) -mixed geominimal surface area and the ( p , q ) -mixed affine surface area are introduced. They are extensions of the L p geominimal surface area and the L p affine surface area, respectively. Affine invariance under unimodular affine transformations and continuity for the ( p , q ) -mixed geominimal surface area and the ( p , q ) -mixed affine surface area are discussed. The existence, the uniqueness and the uniform boundness of the ( p , q ) -mixed Petty bodies are obtained. Affine isoperimetric inequalities for the ( p , q ) -mixed geominimal surface area and the ( p , q ) -mixed affine surface area are established.