Comparison theorem for support functions of hypersurfaces

TL;DR

This paper establishes a comparison theorem for angles and support functions of convex hypersurfaces with bounded normal curvature, leading to a new proof of Blaschke's Rolling Theorem.

Contribution

It introduces a novel angle comparison theorem and support function comparison for convex hypersurfaces with bounded normal curvature, extending classical geometric results.

Findings

Proved an angle comparison theorem for hypersurfaces with bounded normal curvature.

Derived a comparison theorem for support functions of such hypersurfaces.

Provided a new proof of Blaschke's Rolling Theorem.

Abstract

For a convex domain $D$ that is enclosed by the hypersurface $\partial D$ of bounded normal curvature, we prove an angle comparison theorem for angles between $\partial D$ and geodesic rays starting from some fixed point in $D$, and the corresponding angles for hypersurfaces of constant normal curvature. Also, we obtain a comparison theorem for support functions of such surfaces. As a corollary, we present a proof of Blaschke's Rolling Theorem.