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

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, with the formula depending on one function defined on S2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {S}}^2$$\\end{document}. Each such function gives a minimal value r0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0$$\\end{document} and for all r≥r0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge r_0$$\\end{document} 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 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”.