Sharp affine isoperimetric inequalities for the volume decomposition functionals of polytopes

Abstract

The nth power of the volume functional Vn of polytopes P in Rn, according to dimensions of the spaces spanned by any n unit outer normal vectors of P, is decomposed into n homogeneous polynomials of degree n. A set of new sharp affine isoperimetric inequalities for these volume decomposition functionals in R3 are established, which essentially characterize the geometric and algebraic structures of polytopes.