Sommets et normales concourantes des courbes convexes de largeur constante et singularités des hérissons

Abstract

LetC be a convex curve of constant width and of classC 4 + . It is known thatC has at least 6 vertices and its interior contains either a point through which infinitely many normals pass or an open set of points through each of which pass at least 6 normals. If all its vertices are nondegenerate, then: (i)C has exactly 6 vertices if, and only if, its evolute is the boundary of a topological disc through each interior point of which pass at least 6 normals; (ii) ifC has more than 6 vertices, then there exists an open set of points through each of which pass at least 10 normals. The proof: (i) expresses the number of normals passing through a point as a function of the index with respect to the evolute; (ii) relates this index to the number of singularities of the evolute (i.e. of vertices). Furthermore, we give formulas for counting singularities of generic hedgehogs in ℝ2 and ℝ3.