Approximation of Spherical Bodies of Constant Width and Reduced Bodies

TL;DR

This paper extends Blaschke's theorem to spherical geometry, showing bodies of constant width can be approximated by smoother bodies with boundaries made of circular arcs, advancing the understanding of spherical convex bodies.

Contribution

It introduces a spherical version of Blaschke's theorem and provides approximation results for spherical bodies of constant width and reduced bodies.

Findings

Bodies of constant width less than π/2 can be approximated by bodies with circular arc boundaries.

The approximation can be made arbitrarily close in Hausdorff distance.

The results apply to spherical reduced bodies, broadening classical Euclidean convex geometry.

Abstract

We present a spherical version of the theorem of Blaschke that every body of constant width $w < \frac{\pi}{2}$ can be approximated as well as we wish in the sense of the Hausdorff distance by a body of constant width $w$ whose boundary consists only of pieces of circles of radius $w$. This is a special case of our theorem about approximation of spherical reduced bodies.