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

TL;DR

This paper introduces weighted cone-volume functionals for convex polytopes, establishes geometric inequalities, characterizes equality cases, and explores extremal properties of regular polytopes with applications in crystallography and quantum theory.

Contribution

It develops new weighted cone-volume functionals, proves related inequalities, and characterizes extremal configurations, extending classical geometric results to broader contexts.

Findings

Proved geometric inequalities for weighted cone-volume functionals

Characterized conditions for equality in these inequalities

Identified extremal properties of regular polytopes involving $L_p$ surface area

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.