On hypersurfaces of positive reach, alternating Steiner formulae and Hadwiger's Problem

TL;DR

This paper characterizes hypersurfaces with positive reach, establishes new Steiner formulae for them, and applies these results to solve a problem by Hadwiger and analyze the gradient flow of mean breadth.

Contribution

It introduces new characterizations of positive reach hypersurfaces, derives alternating Steiner formulae, and solves Hadwiger's problem with implications for geometric flows.

Findings

Hypersurfaces of positive reach are exactly the $C^{1,1}$ class.

New alternating Steiner formulae are established for such hypersurfaces.

A solution to Hadwiger's problem and long-term existence of mean breadth gradient flow are provided.

Abstract

We give new characterisations of sets of positive reach and show that a closed hypersurface has positive reach if and only if it is of class $C^{1,1}$. These results are then used to prove new alternating Steiner formul{\ae} for hypersurfaces of positive reach. Furthermore, it will turn out that every hypersurface that satisfies an alternating Steiner formula has positive reach. Finally, we provide a new solution to a problem by Hadwiger on convex sets and prove long time existence for the gradient flow of mean breadth.