On a formula for all sets of constant width in 3d

TL;DR

This paper extends a constructive formula from 2D to 3D for bodies of constant width, providing an explicit parametrization based on a function on the sphere, and shows all such bodies can be described this way.

Contribution

It derives a new explicit parametrization formula for 3D bodies of constant width, generalizing previous 2D results and demonstrating its universality for all such bodies.

Findings

Provides a parametrization depending on a function on S^2

Shows all 3D bodies of constant width have this parametrization

Introduces the 'shadow domain' tool for the construction

Abstract

In the recent paper "On a formula for sets of constant width in 2D", Comm. Pure Appl. Anal. 18 (2019), 2117-2131, we gave a constructive formula for all 2d sets of constant width. Based on this result we derive here a formula for the parametrization of the boundary of bodies of constant width in 3 dimensions, depending on one function defined on S^2. Each such function gives a minimal value r_0 and for all r \ge r_0 one finds a body of constant width 2r. Moreover, we show that all bodies of constant width in 3d have such a parametrization. The last result needs a tool that we describe as 'shadow domain' and that is explained in an appendix. The construction is explicit and and offers a parametrization different from the one given by T. Bayen, T. Lachand-Robert and \'E. Oudet, "Analytic parametrization of three-dimensional bodies of constant width" in Arch. Ration. Mech. Anal., 186 (2007), 225-249.