21
$\begingroup$

The standard Reuleaux triangle is not smooth, but the three points of tangential discontinuity can be smoothed as in the figure below (left), from the Wikipedia article. However, it is unclear (to me) from this diagram whether the curve is $C^2$ or $C^\infty$.

Meissner’s tetrahedron is a 3D body of constant width, but it is not smooth, as is evident in the right figure below.

   Constant Width

My question is:

Are there $C^\infty$ constant-width bodies in $\mathbb{R}^d$ (other than the spheres)?

The image of Meissner’s tetrahedron above is taken from the impressive work of Thomas Lachand–Robert and Edouard Oudet, "Bodies of constant width in arbitrary dimension" (Math. Nachr. 280, No. 7, 740-750 (2007); pre-publication PDF here). Here is a link to Wayback Machine.)

I suspect the answer to my question is known, in which case a reference would suffice. Thanks!

Addendum. Thanks to the knowledgeable (and rapid!) answers by Gerry, Anton, and Andrey, my question is completely answered—I am grateful!!

$\endgroup$
5
  • 7
    $\begingroup$ I trust you mean, other than the sphere. $\endgroup$ Feb 3, 2011 at 22:40
  • 1
    $\begingroup$ @Gerry: Yes, that's what I meant, thanks; updated the question. $\endgroup$ Feb 3, 2011 at 22:48
  • 1
    $\begingroup$ You ask whether the curve on the diagram is smooth. It isn't: it's made from six circle arcs. $\endgroup$ Feb 4, 2011 at 10:25
  • 1
    $\begingroup$ Thanks, Zsbán. So I guess just $C^1$? $\endgroup$ Feb 4, 2011 at 10:45
  • 1
    $\begingroup$ The construction by three non-incident straight lines & circle arcs, is the simplest, but not the unique way to make figures with constant width; the same works of course with 4 or more lines. The continuous version as well, and allows smooth objects, in any dimension $d$ (less immediate for $d>2$). $\endgroup$ Dec 27, 2022 at 13:09

6 Answers 6

31
$\begingroup$

Fillmore showed that there are sets of constant width in $\mathbb R^d$ with analytic boundaries which have a trivial symmetry group (so these are very different from spheres; see "Symmetries of surfaces of constant width", J. Differential Geom., Vol 3, (1969), pp. 103-110).

Moreover, the set of bodies of constant width with analytic boundaries is dense in the space of all convex bodies of constant width in $\mathbb R^d$ with respect to the Hausdorff metric (see e.g. "Smooth approximation of convex bodies" by Schneider).

$\endgroup$
3
  • 1
    $\begingroup$ In the second paragraph, a "of constant width" is missing; it is implicit, but it disturbed me for a second. $\endgroup$ Feb 11, 2011 at 12:11
  • 6
    $\begingroup$ For an ignoramus like me, these results are quite impressive and highly counter-intuitive. $\endgroup$
    – Olivier
    Dec 11, 2011 at 14:19
  • 2
    $\begingroup$ I took a look at the Fillmore paper, and just before his Corollary to Theorem 2 -- which reads "Corollary. There exists an analytic hypersurface of constant width in E^n having the same group of symmetries as a regular n-simplex." -- he writes "If we imitate the construction of a Reuleux triangle . . .. Thus:" This seems to imply that he is assuming that [the intersection of four balls in 3-space, centered at the vertices of a regular tetrahedron and each with radius = the side-length of the tetrahedron] is a body of constant width. But this is known to be false. $\endgroup$ Jan 11, 2013 at 4:50
26
$\begingroup$

Take any odd $C^\infty$-function $f$ on the sphere. Consider the convex set $$R_\epsilon=\{\,x\in\mathbb R^n\mid\langle x,u\rangle\le 1+\epsilon{\cdot}f(u)\ \ \text{for any}\ \ u\in\mathbb{S}^{n-1}\,\}.$$ Note that for all sufficiently small $\epsilon>0$, $R_\epsilon$ is a smooth body of constant width.

$\endgroup$
16
$\begingroup$

Jay P Fillmore, Symmetries of surfaces of constant width, J Differential Geometry 5 (1969) 103-110, says: the curve $$x_1=h\cos\theta-{dh\over d\theta}\sin\theta,\qquad x_2=h\sin\theta+{dh\over d\theta}\cos\theta$$ where $h=a+b\cos3\theta$, $0\lt8b\lt a$, is analytic and of constant width. If we rotate this curve in Euclidean space ${\bf E}^n$ about an $(n-2)$-dimensional axis perpendicular to the line $\theta=0$, we obtain an analytic surface, not a sphere, of constant width in ${\bf E}^n$.

The paper may be available at https://mathweb.ucsd.edu/~fillmore/papers/a.html, https://doi.org/10.4310/jdg/1214428822, https://projecteuclid.org/euclid.jdg/1214428822

$\endgroup$
8
$\begingroup$

Take any odd smooth function h on the unit (d-1)-sphere and take a constant r>0 large enough to ensure that h+r is the support function of a convex body K

(the condition for h+r to be the support function of a smooth convex body whose boundary has positive Gaussian curvature is that the eigenvalues of Hess(h)+(h+r).Id be positive).

This convex body K is of constant width 2r.

Moreover, any smooth convex body with constant width 2r whose boundary has positive Gaussian curvature can be constructed in this way :

If S is a closed convex hypersurface of constant width 2r, then S is the sum of a sphere of radius r with a "projective hedgehog" H whose support function h is the odd part of the support function of S (and which can be regarded as the locus of the middles of S's diameters)." ;

See for instance:

Y. Martinez-Maure, Arch. Math., Vol. 67, 156-163 (1996), page 157.

$\endgroup$
7
$\begingroup$

This relevant paper was just released:

Howard Resnikoff. "On Curves and Surfaces of Constant Width." Apr. 2015. (arXiv abstract.)

"...we provide a Fourier series-based construction that produces arbitrarily many new surfaces of constant width."


      ![ConstWidth][1]
$\endgroup$
1
  • 1
    $\begingroup$ Also related: Isaac Arelio, Luis Montejano, Deborah Oliveros. "Peabodies of Constant Width." arXiv abs. $\endgroup$ Jul 14, 2021 at 13:55
6
$\begingroup$

Michael Kallay characterized the set of all planar sets with a given width functions: See M. Kallay, Reconstruction of a plane convex body from the curvature of its boundary. Israel J. Math. 17 (1974), 149–161. and M. Kallay, The extreme bodies in the set of plane convex bodies with a given width function. Israel J. Math. 22 (1975), no. 3-4, 203–207.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.