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

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.