Convex Polytopes, Dihedral Angles, Mean Curvature and Scalar Curvature

TL;DR

This paper develops a method to approximate convex polytope boundaries with smooth hypersurfaces of positive mean curvature, establishing lower bounds on dihedral angles through geometric relations involving scalar and mean curvatures.

Contribution

It introduces a novel approximation technique for convex polytopes and derives new bounds on dihedral angles using scalar and mean curvature relations.

Findings

Bound on dihedral angles of convex polytopes

Approximation of polytope boundaries by smooth hypersurfaces

Relation between scalar curvature and mean curvature

Abstract

We approximate boundaries of convex polytopes by smooth hypersurfaces $Y=Y_\varepsilon$ with {\it positive mean curvatures} and, by using basic geometric relations between the scalar curvatures of Riemannin manifolds and the mean curvatures of their boundaries, establish {\it lower bound on the dihedral angles} of these polytopes.