Analytic Parametrization of Three-Dimensional Bodies of Constant Width

Abstract

We present a complete analytic parametrization of constant three-dimensional width bodies based on the median surface: more precisely, we define a bijection between spaces of functions and constant width bodies. We compute simple geometrical quantities like the volume and the surface area in terms of those functions. As a corollary we give a new algebraic proof of Blaschke’s formula. Finally, we derive weak optimality conditions for convex bodies which minimize the volume among constant width bodies.