Weak convexity does not imply convexity for curves in P n , n>2
[La convexité faible n'implique pas la convexité des courbes dans P n , n>2]
Comptes Rendus. Mathématique, Tome 335 (2002) no. 1, pp. 47-52.

Une courbe lisse fermée dans P n est appelée convexe si chaque hyperplan l'intersecte en au plus n points, compte tenu des multiplicités. Une courbe convexe n'a pas d'aplatissement et son hyperplan osculateur ne l'intersecte qu'au point d'osculation. Une courbe fermée dans P 2 est convexe si et seulement si elle a ces deux propriétés. En réponse à une question de V.I. Arnol'd ([2,3] et [4]), nous montrons que pour n>2, ces deux propriétés n'impliquent pas la convexité des courbes fermées dans P n .

A smooth closed curve in P n is called convex if any hyperplane intersects it in at most n points, taking multiplicities into account. A convex curve has no flattening and its osculating hyperplane intersects it only at the point of osculation. A closed curve in P 2 (in 2 ) is convex if and only if it has these two properties. Answering a question of V.I. Arnol'd ([2,3] and [4]), we show that, for n>2, these two properties do not imply the convexity of closed curves in P n .

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DOI : 10.1016/S1631-073X(02)02435-4
Uribe-Vargas, Ricardo 1

1 Université Paris 7, Équipe géométrie et dynamique, UFR de Math., case 7012, 2, place Jussieu, 75005 Paris, France
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Uribe-Vargas, Ricardo. Weak convexity does not imply convexity for curves in $ \mathbb{R}P^{n}$, n>2. Comptes Rendus. Mathématique, Tome 335 (2002) no. 1, pp. 47-52. doi : 10.1016/S1631-073X(02)02435-4. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02435-4/

[1] Anisov, S.S. Convex curves in P n , Proc. Steklov Math. Inst., Volume 221 (1998), pp. 3-39

[2] Arnol'd, V.I. On the number of flattening points of space curves, Amer. Math. Soc. Trans. Ser., Volume 171 (1995), pp. 11-22

[3] Arnol'd, V.I. Topological problems of the theory of wave propagation, Russian Math. Surveys, Volume 51 (1996) no. 1, pp. 1-47

[4] Arnol'd, V.I. Problem 1994–15, Arnol'd's Problems Book, Phasis, 1999 (in Russian). English edition to appear

[5] Barner, M. Über die Mindestanzahl stationärer Schmiegeebenen bei geschlossenen Streng-Konvexen Raumkurven, Abh. Math. Sem. Univ. Hamburg, Volume 20 (1956), pp. 196-215

[6] Sedykh, V.D. The theorem about four vertices of a convex space curve, Funct. Anal. Appl., Volume 26 (1992) no. 1, pp. 28-32

[7] Uribe-Vargas, R. On the higher dimensional four-vertex theorem, C. R. Acad. Sci. Paris, Série I, Volume 321 (1995), pp. 1353-1358

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