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

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, geominimal surface area and the L, 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. (C) 2019 Published by Elsevier Inc.