most recent 30 from http://mathoverflow.net 2013-05-18T23:33:42Z http://mathoverflow.net/feeds/user/22068 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90862/what-are-some-examples-of-ingenious-unexpected-constructions/90890#90890 Yves 2012-03-11T09:19:06Z 2012-03-11T09:19:06Z

<p>Disproving an old A. D. Alexandrov's hypothesis on a characterization of the 2-sphere, Y. Martinez-Maure constructed a curious object in Euclidean 3-space $R^3$: a closed oriented surface M that has the following properties:</p> <p>1) M is a smooth saddle surface at each of its points (except for four points called ‘horns’): every point x of M distinct from a horn is a hyperbolic point of M (the Gaussian curvature of M is negative at x).</p> <p>2) M possesses (as smooth strictly convex surfaces) a well-defined smooth support function on $S^2$. In particular, its Gauss map $M → S^2$ is injective.</p> <p>See :</p> <p>M G Knyazeva and Gayane Yu Panina, Russian Mathematical Surveys Volume 63, 2008.</p> <p>and/or</p> <p>Y. Martinez-Maure, Contre-exemple à une caractérisation conjecturée de la sphère, Comptes Rendus de l'Académie des Sciences de Paris 332, Série I, 2001, 41-44.</p>