Abstract: In this talk, I will introduce our very recent work on the geometry of polytopes. We prove that the nth power of the volume functional Vn of polytopes P in Rn, according to dimensions of the spaces spanned by any n outer normal unit vectors of P, is naturally decomposed into n homogeneous polynomials with degree n. Several new sharp affine isoperimetric inequal-ities for these functionals are established, which essentially characterize the geometric and algebraic structures of polytopes.
