Vertices and normals passing trough a point of convex curves of constant width and hedgehogs singularities. (Sommets et normales concourantes des courbes convexes de largeur constante et singularités des hérissons.) (French) Zbl 1025.52004

It is known that planar convex curves of constant width, of class \(C^4\) and with strictly positive curvature have at least 6 vertices [G. D. Chakerian and H. Groemer, Convexity and its applications, Collect. Surv., 49-96 (1983; Zbl 0518.52002)]. It is also known that these curves either contain a point through which infinitely many normals pass or an open set of points through each of which pass at least 6 normals [Y. Martinez-Maure, Publ. Mat., Barc. 44, 237-255 (2000; Zbl 0974.53003)].
The author proves in this paper that both properties are closely related. The main result states that if with the above assumptions all its vertices are nondegenerate, then (i) the curve 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) if the curve 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).
Besides the author gives some formulae for counting singularities of generic hedgehogs in euclidean space of dimensions 2 and 3.


52A10 Convex sets in \(2\) dimensions (including convex curves)
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
53A05 Surfaces in Euclidean and related spaces
57R45 Singularities of differentiable mappings in differential topology
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[1] V. I. Arnold, Critical points of smooth functions. Proc. Int. Cong. Math., Vancouver 1974, vol. 1, 19–39.
[2] T. Banchoff etR. Thom, Sur les points paraboliques des surfaces: erratum et compléments. C. R. Acad. Sci. Paris291, 503–505 (1980). · Zbl 0463.58007
[3] M. Berger etB. Gostiaux, Géométrie différentielle: Variétés, Courbes et Surfaces. Presses Univ. France, 1987.
[4] G. D. Chakerian etH. Groemer, Convex bodies of constant width. Dans: Convexity and its applications, Collect. Surv., 49–96 (1983).
[5] D. Hilbert etS. Cohn-Vossen, Geometry and the imagination. New York 1952. Ouvrage original en allemand: Anschauliche Geometrie, Berlin 1932.
[6] E. Heil, Existenz eines 6-Normalenpunktes in einem konvexen Körper. Arch. Math.32, 412–416 (1979). · Zbl 0404.52006
[7] E. Heil, Korrektur zu: ”Existenz eines 6-Normalenpunktes in einem konvexen Körper”. Arch. Math.33, 496 (1979–80). · Zbl 0426.52001
[8] Y. Martinez-Maure, Sur les hérissons projectifs (enveloppes paramétrées par leur application de Gauss). Bull. Sci. Math.121, 585–601 (1997). · Zbl 0893.53002
[9] Y. Martinez-Maure, Indice d’un hérisson: étude et applications. Publ. Mat.44, 237–255 (2000). · Zbl 0974.53003
[10] R. Langevin, G. Levitt etH. Rosenberg, Hérissons et multihérissons (enveloppes paramétrées par leur application de Gauss). Singularities. Banach Center Publ.20, 245–253, Warsaw 1988.
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