On extreme constant width bodies in $\mathbb{R}^3$

TL;DR

This paper investigates extreme shapes within the family of convex constant width bodies in three-dimensional space, identifying specific classes like Meissner polyhedra and rotated Reuleaux polygons as extreme, and proposing a conjecture for their complete characterization.

Contribution

It demonstrates that Meissner polyhedra and certain rotated Reuleaux polygons are extreme constant width bodies, and proposes a conjecture for a full classification of all such extreme shapes.

Findings

Meissner polyhedra are extreme constant width bodies.

Rotations of symmetric Reuleaux polygons are extreme.

A conjecture for characterizing all extreme constant width bodies is proposed.

Abstract

We consider the family of constant width bodies in $\mathbb{R}^3$ which is convex under Minkowski addition. Extreme shapes cannot be expressed as a nontrivial convex combination of other constant width bodies. We show that each Meissner polyhedra is extreme. We also explain that each constant width body obtained by rotating a symmetric Reuleaux polygon about its axis of symmetry is extreme. In addition, we conjecture a general characterization of all extreme constant width shapes.