On a Formula for All Sets of Constant Width in 3D

• Published in: The Journal of geometric analysis Vol. 34; no. 7
• Main Authors: Kawohl, Bernd, Sweers, Guido
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2024; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Global Analysis and Analysis on Manifolds; Mathematics; Mathematics and Statistics; Parameterization; Three dimensional analysis; Three dimensional bodies; Convex geometry; 52A15; Constant width; 3-dimensional
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-024-01622-9

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, with the formula depending on one function defined on S 2 . Each such function gives a minimal value r 0 and for all r ≥ r 0 one finds a body of constant width 2 r . 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 which is explained in an appendix. The construction is explicit and offers a parametrization different from the one given by T. Bayen, T. Lachand-Robert and É. Oudet in “Analytic parametrization of three-dimensional bodies of constant width. Arch. Ration. Mech. Anal., 186 (2007), 225–249”.