Existenz eines 6-Normalenpunktes in einem konvexen K�rper

Heil, Erhard

doi:10.1007/bf01238519

Cited by 13 publications

(13 citation statements)

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“…Now we show that the results of Section 1 are still true if the smoothness condition (dK E C2) is dropped. (This was also claimed in [7], but the proof given there is valid only for dK E C2.) For an arbitrary convex body K c En (n 2 3), a normal is a straight line orthogonal to a supporting hyperplane at one of its points of support.…”

Section: The Non-smcoth Casesupporting

confidence: 57%

“…The focal surface 9 of the ellipsoid in E3 was studied by Caley and others. It divides E3 into regions with two, four, or six normals through each point; see [7] for an illustration and more details. The situation is similar in the general case (dK E C2) as we shall see.…”

Section: The Set Of Non-focal Pointsmentioning

confidence: 99%

“…More can be said using the theory of degree of mappings. Some details can be found in [7]. The normal bundle is orientable and the degree of CP is even, namely, 1 -(-1)".…”

Section: The Set Of Non-focal Pointsmentioning

confidence: 99%

“…The number of normals passing through a point p 4 9 is even and is constant on the open sets which are the connected components of the complement of9 (dK E Cz).For sets of constant width the number of normals through a non-focal point is congruent to 2 modulo 4. For details see[7].…”

mentioning

confidence: 99%

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Concurrent Normals and Critical Points Under Weak Smoothness Assumptions

Heil

1985

Annals of the New York Academy of Sciences

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It is shown that in any convex body K there is a point which is met by six normals. Essential tools are Morse's relations for the number of critical points. In Section 1 we assume for simplicity that the boundary of K is of class C2. In Section 3 we show that convexity itself provides enough smoothness. Hereby an earlier paper [7] is corrected where it was erroneously claimed that the proof given there was valid for arbitrary convex bodies, but in fact only worked for bodies of class C2. The proof given here is also simpler since it does not need the theory of degree of mappings. In Section 2 we discuss what further information is given by the degree. In Section 4 we list some open problems.

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Section: The Non-smcoth Casesupporting

confidence: 57%

Section: The Set Of Non-focal Pointsmentioning

confidence: 99%

“…More can be said using the theory of degree of mappings. Some details can be found in [7]. The normal bundle is orientable and the degree of CP is even, namely, 1 -(-1)".…”

Section: The Set Of Non-focal Pointsmentioning

confidence: 99%

mentioning

confidence: 99%

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Concurrent Normals and Critical Points Under Weak Smoothness Assumptions

Heil

1985

Annals of the New York Academy of Sciences

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“…Nous démontrerons ensuite une propriété remarquable des normalesà un corps convexe plan de largeur constante et de classe C 2 + : il existe un point par lequel il passe une infinité de normales ou un ouvert formé de points par lesquels il passe au moins 6 normales. Notre résultat, qui sera plus général, correspondà celuiétabli par E. Heil dans R 3 : un corps convexe de largeur constante de R 3 contient un point par lequel il passe une infinité de normales ou un ouvert formé de points par lesquels il passe au moins 10 normales (voir [3] et [4]).…”

Section: Introductionunclassified