The mixed L_p affine surface areas for functions

Abstract

In this paper, we propose a definition of a general mixed L_p Affine surface area, −n ≠ p ∈ ℝ, for multiple functions. Our definition is different from and is “dual” to the one in [11] by Caglar and Ye. In particular, our definition makes it possible to establish an integral formula for the general mixed L_p Affine surface area of multiple functions (see Theorem 3.1 for more precise statements). Properties of the newly introduced functional are proved such as affine invariance, and related affine isoperimetric inequalities are proved.