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Weighted cone-volume functionals are introduced for the convex polytopes in$\mathbb{R}^n$. For these functionals, geometric inequalities are proved andthe equality conditions are characterized. A variety of corollaries arederived, including extremal properties of the regular polytopes involving the$L_p$ surface area. Some applications to crystallography and quantum theory arealso presented.

Extremal arrangements of points on the sphere for weighted cone-volume functionals