Are there smooth bodies of constant width? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:41:23Z http://mathoverflow.net/feeds/question/54252 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54252/are-there-smooth-bodies-of-constant-width Are there smooth bodies of constant width? Joseph O'Rourke 2011-02-03T22:32:38Z 2011-12-11T14:12:37Z <p>The standard Reuleaux triangle is not smooth, but the three points of tangential discontinuity can be smoothed as in the figure below (left), from <a href="http://en.wikipedia.org/wiki/Curve_of_constant_width" rel="nofollow">the Wikipedia article</a>. However, it is unclear (to me) from this diagram whether the curve is $C^2$ or $C^\infty$.</p> <p><em>Meissner’s tetrahedron</em> is a 3D body of constant width, but it is not smooth, as is evident in the right figure below.</p> <p><br /> &nbsp;&nbsp;&nbsp;<img src="http://people.csail.mit.edu/~orourke/MathOverflow/ConstantWidth.jpg" alt="Constant Width"> <br /></p> <p>My question is:</p> <blockquote> <p>Are there $C^\infty$ constant-width bodies in $\mathbb{R}^d$ (other than the spheres)?</p> </blockquote> <p>The image of Meissner’s tetrahedron above is taken from <a href="http://www.lama.univ-savoie.fr/~lachand/Spheroforms.html" rel="nofollow">the impressive work</a> of Thomas Lachand–Robert and Edouard Oudet, "Bodies of constant width in arbitrary dimension" (<em>Math. Nachr.</em> 280, No. 7, 740-750 (2007); <a href="http://www.lama.univ-savoie.fr/~lachand/pdfs/spheroforms.pdf" rel="nofollow">pre-publication PDF here</a>).</p> <p>I suspect the answer to my question is known, in which case a reference would suffice. Thanks!</p> <p><b>Addendum.</b> Thanks to the knowledgeable (and rapid!) answers by Gerry, Anton, and Andrey, my question is completely answered&mdash;I am grateful!! </p> http://mathoverflow.net/questions/54252/are-there-smooth-bodies-of-constant-width/54254#54254 Answer by Gerry Myerson for Are there smooth bodies of constant width? Gerry Myerson 2011-02-03T22:46:06Z 2011-02-03T23:19:28Z <p>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 ${\bf E}^n$. </p> <p>The paper may be available at <a href="http://www.intlpress.com/JDG/archive/1969/3-1&amp;2-103.pdf" rel="nofollow">http://www.intlpress.com/JDG/archive/1969/3-1&amp;2-103.pdf</a></p> http://mathoverflow.net/questions/54252/are-there-smooth-bodies-of-constant-width/54258#54258 Answer by Anton Petrunin for Are there smooth bodies of constant width? Anton Petrunin 2011-02-03T23:20:33Z 2011-02-03T23:20:33Z <p>Take any odd $C^\infty$-function $f$ on the sphere. Consider convex set $$R_\epsilon=\{\,x\in\mathbb R^n\mid\langle x,u\rangle\le 1+\epsilon{\cdot}f(u)\ \ \text{for any}\ \ u\in\mathbb{S}^{n-1}\,\}.$$ Clearly for all sufficiently small $\epsilon>0$, $R_\epsilon$ is a smooth body of constant width.</p> http://mathoverflow.net/questions/54252/are-there-smooth-bodies-of-constant-width/54260#54260 Answer by Andrey Rekalo for Are there smooth bodies of constant width? Andrey Rekalo 2011-02-03T23:23:39Z 2011-02-11T12:16:07Z <p>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 <a href="http://www.projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.jdg/1214428822" rel="nofollow">"Symmetries of surfaces of constant width"</a>, <em>J. Differential Geom.</em>, Vol 3, (1969), pp. 103-110).</p> <p>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. <a href="http://www.springerlink.com/content/8072555602p48411/" rel="nofollow">"Smooth approximation of convex bodies"</a> by Schneider). </p> http://mathoverflow.net/questions/54252/are-there-smooth-bodies-of-constant-width/55112#55112 Answer by Gil Kalai for Are there smooth bodies of constant width? Gil Kalai 2011-02-11T11:13:18Z 2011-02-11T11:13:18Z <p>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.</p> http://mathoverflow.net/questions/54252/are-there-smooth-bodies-of-constant-width/83173#83173 Answer by Guillaume for Are there smooth bodies of constant width? Guillaume 2011-12-11T10:44:02Z 2011-12-11T10:44:02Z <p>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</p> <p>(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).</p> <p>This convex body K is of constant width 2r.</p> <p>Moreover, any smooth convex body with constant width 2r whose boundary has positive Gaussian curvature can be constructed in this way :</p> <p>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)." ;</p> <p>See for instance:</p> <p>Y. Martinez-Maure, Arch. Math., Vol. 67, 156-163 (1996), page 157.</p>