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

Bernd Kawohl, Guido Sweers

Published 2023-07-06Version 1

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 and a large enough constant. 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. Our formula is more explicit than the result 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.