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.
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Liu et al. (2024) studied this question.
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