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Comptes Rendus Mathématique
Volume 335, n° 1
pages 47-52 (2002)
Doi : S1631-073X(02)02435-4
Received : 23 April 2002 ;  accepted : 3 May 2002
Weak convexity does not imply convexity for curves in RPn, n >2
La convexité faible n'implique pas la convexité des courbes dans RPn, n >2
 

Ricardo Uribe-Vargas
Université Paris 7, Équipe géométrie et dynamique, UFR de Math., case 7012, 2, place Jussieu, 75005 Paris, France 

Note presented by Vladimir Arnol'd

Abstract

A smooth closed curve in RPn 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 RP2 (in R2) 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 RPn . To cite this article: R. Uribe-Vargas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 47-52.

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Résumé

Une courbe lisse fermée dans RPn 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 RP2 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 RPn . Pour citer cet article : R. Uribe-Vargas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 47-52.

The full text of this article is available in PDF format.


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