[La convexité faible n'implique pas la convexité des courbes dans , n>2]
Une courbe lisse fermée dans 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 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 .
A smooth closed curve in 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 (in ) 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 .
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@article{CRMATH_2002__335_1_47_0,
author = {Uribe-Vargas, Ricardo},
title = {Weak convexity does not imply convexity for curves in~$ \mathbb{R}P^{n}$, \protect\emph{n}>2},
journal = {Comptes Rendus. Math\'ematique},
pages = {47--52},
publisher = {Elsevier},
volume = {335},
number = {1},
year = {2002},
doi = {10.1016/S1631-073X(02)02435-4},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02435-4/}
}
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AU - Uribe-Vargas, Ricardo
TI - Weak convexity does not imply convexity for curves in $ \mathbb{R}P^{n}$, n>2
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PY - 2002
SP - 47
EP - 52
VL - 335
IS - 1
PB - Elsevier
UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02435-4/
DO - 10.1016/S1631-073X(02)02435-4
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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/
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