Recursion formulas, concentration polytope, and sharp affine isoperimetric inequalities for the volume decomposition functionals

New sharp affine isoperimetric inequalities for volume decomposition functionals X₂ and X₃ in Rⁿ are established. To attack these extremal problems, we find the recursion formulas of volume decomposition functionals, then introduce concentration polytope and characterize its geometric structure, especially its vertices and 1-dimensional faces. The concentration polytope intuitively reinterprets the discrete logarithmic Minkowski problem.