The Fillmore paper (1968!) looks fascinating, but hard work for those not conversant with the jargon of spherical harmonic functions. The reference below does at least explain (pp. 8--9) what a "(Minkowski) support function" means. Referring back to Dan's original enquiry: according to p. 260 of Appendix I in V.V. Buldygin, A.B. Kharazishvili "Geometric Aspects of Probability Theory and Mathematical Statistics" --- for _any_ subset of Euclidean n-space, there exists a constant width set containing it, and having the same diameter (ie. width). Evidently from Fillmore's paper, the construction may fail to preserve the symmetry group --- of course, for any regular polytope, a sphere suffices. In fact, just at the moment, I am having difficulty imagining a set which is _not_ contained in a sphere of the same diameter ... Fred Lunnon On 1/9/13, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks, Fred! There's a paper by Jay Fillmore linked from the Math Overflow post you linked to, and the corollary to Fillmore's Theorem 2 is this:
"There exists an analytic hypersurface of constant width in E^n having the same group of symmetries as a regular n-simplex.
Which is just what I was wondering (though the real analytic differentiability is a bonus).
--Dan
I wrote:
<< QUESTION: Has anyone proved that a 3D shape of constant width *cannot* have the symmetry of the tetrahedron? (Meaning no additional symmetries, either.)
On 2013-01-08, at 7:16 PM, Fred lunnon wrote:
[Thread mysteriously split into two ...]
Consider the following construction of a continuously differentiable surface, with tetrahedral symmetry: (0) start from a regular tetrahedron of circumradius 1; (1) balloon each face out to a spherical cap of radius 1+r, bounded where it meets each of the other 3 faces extended; (2) balloon each vertex out to a cap of radius r, bounded similarly; (3) balloon each edge out the the unique Dupin cyclide tangent to all 4 adjacent spherical boundaries.
Does this have constant width? Meissner's surface seems to be the case r = 0, which is not differentiable.
A couple of recent references turned up via Google ---
mathoverflow.net/.../are-there-smooth-bodies-of-constant-width
www.mi.uni-koeln.de/mi/Forschung/Kawohl/kawohl/pub100.pdf
One response under the first thread suggests a method of constructing arbitrary _analytic_ surfaces of constant width, but is so cryptic that I can make neither head nor tail of it.
Fred Lunnon