Recursion formulas, concentration polytope, and sharp affine
isoperimetric inequalities for the volume decomposition functionals
Recursion formulas, concentration polytope, and sharp affine
isoperimetric inequalities for the volume decomposition functionals
New sharp affine isoperimetric inequalities for volume decomposition functionals $X_{2}$ and $X_{3}$ in $\mathbb{R}^n$ 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 …