Come to think of it, Fillmore's Corollary quoted below is preceded by this remark (for readability I'll requote the Corollary, too): ----- If we imitate the construction of a Reuleux triangle in n-dimension, a parallel hypersurface at constant distance from this hypersurface is of class C1 and has the symmetry group (a finite group) of the regular n-simplex. Thus: Corollary. There exists an analytic hypersurface of constant width in E^n having the same group of symmetries as a regular n-simplex. ----- OOPS. The Reuleaux tetrahedron, for instance, is *not* of constant width. (It's also spelled with an "a".) So Fillmore's Corollary is in serious doubt!!! In any case, the argument in his paper includes a crucial mistake. In fact, according to Wikipedia: ----- The Reuleaux tetrahedron is the intersection of four spheres of radius s centered at the vertices of a regular tetrahedron with side length s. The sphere through each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron. The Reuleaux tetrahedron has the same face structure as a regular tetrahedron, but with curved faces: four vertices, and four curved faces, connected by six circular-arc edges. This shape is defined and named by analogy to the Reuleaux triangle, a two-dimensional curve of constant width. One can find repeated claims in the mathematical literature that the Reuleaux tetrahedron is analogously a surface of constant width, but it is not true: the two midpoints of opposite edge arcs are separated by a larger distance, (sqrt(3) - sqrt(2)/2)s. ----- So: I still don't know if there's a surface of constant width in 3-space having the symmetry group of the tetrahedron. --Dan On 2013-01-09, at 1:51 PM, Fred lunnon wrote:
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
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