On Extreme Constant Width Bodies in $$\mathbb {R}^3$$

Abstract We consider the family of constant width bodies in $$\mathbb {R}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> 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 polyhedron is extreme. We also explain that each constant width body obtained by rotating a symmetric Reuleaux polygon about its line of symmetry is extreme. In addition, we conjecture a general characterization of all extreme constant width shapes.