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

Abstract

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

arXiv comment15 pages, 1 figure; to appear in Discrete Mathematics