The mixed <i>Lp</i> 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 denfiition is different from and is "dual" to the one in [11] by Caglar and Ye. In particular, our denfiition 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.

Mathematics Subject Classification (2010): 52A20, 53A15.

Keywords: Affine isoperimetric inequality, affine surface area, functional inequality, L_p affine surface area, L_p geominimal surface area, Blaschke-Santaló inequality