User guillaume - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T22:22:03Z http://mathoverflow.net/feeds/user/19873 http://www.creativecommons.org/licenses/by-nc/2.5/rdf 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>