A matroid polytope approach to sharp affine isoperimetric inequalities for volume decomposition functionals

New sharp affine isoperimetric inequalities for volume decomposition functionals $X_2$ and $X_3$ in ${\mathbb{R}}^n$ are established. To fulfil this task, we prove the recursion formulas for volume decomposition functionals and find out the connection between the domains of these functionals and matroid polytopes. Applications of matroid theory to convex geometry are presented.