ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 51, 52, 53 Sedykh, Vyacheslav D., Gubkin Academy of Oil and Gas, Moscow, Russia: On classes of convex curves in projective spaces. A smooth (C^{∞}-embedded) closed curve in ℝP^{n+1} is called convex if every hyperplane ℝP^{n} intersects it at most at n+1 points (taking the multiplicities into account). The curve is called convex by Barner if for every its n points (which are not necessary geometrically-different) there exists a hyperplane through these points that does not intersect the curve anymore. A curve in ℝP^{n+1} having a convex projection to a hyperplane ℝP^{n} from a point (outside of the curve) is convex by Barner. The natural question (V. Arnold, 1996): what is the relation between these two classes of curves? The answer is the following: Theorem. There is an open set (in C^{∞}-topology) of smooth closed curves in ℝP^{n+1}, which are convex by Barner and have no convex projections to any hyperplane.