##
**A global theory of flexes of periodic functions.**
*(English)*
Zbl 1066.51007

H. Kneser’s proof of the 4-vertex theorem and a proof by S. Mukhopadhyaya of the theorem on sextactic points of an oval lead the authors to the concept of intrinsic systems. An intrinsic system \(f\) is defined in the following way: If \(p_1,\dots, p_n\in S^1\) then \(f(p_1,\dots, p_n)\) is a map from \(S^1\) into the set of even numbers which obeys several axioms. In the case of the 4-vertex theorem for a closed curve \(C\) in the plane, \(f(p)\) is defined as follows: Consider the greatest circle enclosed by \(C\) and touching \(C\) at \(p\). Then \(f(p)q\) is the multiplicity with which the circle meets \(C\) at \(q\), and is zero if it does not meet \(C\). For an example with \(n= 2\) let \(C\) be a strictly convex curve. Consider the greatest ellipse enclosed by \(C\) and touching \(C\) at \(p_1\) and \(p_2\). Then define \(f(p_1,p_2)\) similarly.

The authors work with intrinsic systems axiomatically. From their results they gain a strengthened form of the 4-vertex theorem: There are at least 4 “clean” vertices. i.e. vertices with osculating circles which do not meet \(C\) again. The theorem on sextactic points is strengthened similarly. In fact the authors also treat the case that \(C\) or \(S^1\) is replaced by a subarc. This requires considerable effort. As already mentioned the authors use ideas of H. Kneser and S. Mukhopadhyaya, but also from the geometry of orders (0. Haupt’s contraction lemma), from S. B. Jackson and from G. Nöbeling.

The generalization referred to in the title is described in the authors’ abstract: “For a real valued periodic smooth function \(u\) on \(\mathbb{R}\), \(n\geq 0\), one defines the osculating polynomial \(\varphi_s\) (of order \(2n+ 1\)) at a point \(s\in\mathbb{R}\) to be the unique trigonometric polynomial of degree \(n\), whose value and first \(2n\) derivatives at s coincide with those of \(u\) at \(s\). We will say that a point \(s\) is a clean maximal flex (resp. clean minimal flex) of the function \(u\) on \(S^1\) if and only if \(\varphi_s\geq u\) (resp. \(\varphi_s\leq u\)) and the preimage \((\varphi- u)^{-1}(0)\) is connected. We prove that any smooth periodic function \(u\) has at least \(n+ 1\) clean maximal flexes of order \(2n+ 1\) and at least \(n+ 1\) clean minimal flexes of order \(2n+ 1\).”

Furthermore, the spaces of trigonometric polynomials are replaced by more general Chebyshev spaces. A Chebyshev space of order \(2n+ 1\) has dimension at least \(2n+ 1\) and consists of \(2\pi\)-periodic \(C^{2n}\)-functions with at most \(2n\) zeros. The basic properties of Chebyshev spaces are described in an appendix.

The authors work with intrinsic systems axiomatically. From their results they gain a strengthened form of the 4-vertex theorem: There are at least 4 “clean” vertices. i.e. vertices with osculating circles which do not meet \(C\) again. The theorem on sextactic points is strengthened similarly. In fact the authors also treat the case that \(C\) or \(S^1\) is replaced by a subarc. This requires considerable effort. As already mentioned the authors use ideas of H. Kneser and S. Mukhopadhyaya, but also from the geometry of orders (0. Haupt’s contraction lemma), from S. B. Jackson and from G. Nöbeling.

The generalization referred to in the title is described in the authors’ abstract: “For a real valued periodic smooth function \(u\) on \(\mathbb{R}\), \(n\geq 0\), one defines the osculating polynomial \(\varphi_s\) (of order \(2n+ 1\)) at a point \(s\in\mathbb{R}\) to be the unique trigonometric polynomial of degree \(n\), whose value and first \(2n\) derivatives at s coincide with those of \(u\) at \(s\). We will say that a point \(s\) is a clean maximal flex (resp. clean minimal flex) of the function \(u\) on \(S^1\) if and only if \(\varphi_s\geq u\) (resp. \(\varphi_s\leq u\)) and the preimage \((\varphi- u)^{-1}(0)\) is connected. We prove that any smooth periodic function \(u\) has at least \(n+ 1\) clean maximal flexes of order \(2n+ 1\) and at least \(n+ 1\) clean minimal flexes of order \(2n+ 1\).”

Furthermore, the spaces of trigonometric polynomials are replaced by more general Chebyshev spaces. A Chebyshev space of order \(2n+ 1\) has dimension at least \(2n+ 1\) and consists of \(2\pi\)-periodic \(C^{2n}\)-functions with at most \(2n\) zeros. The basic properties of Chebyshev spaces are described in an appendix.

Reviewer: Erhard Heil (Darmstadt)

### MSC:

51L15 | \(n\)-vertex theorems via direct methods |

53C75 | Geometric orders, order geometry |

53A15 | Affine differential geometry |

42A05 | Trigonometric polynomials, inequalities, extremal problems |

### Keywords:

4-vertex theorem; flex; intrinsic systems; sextactic points; contraction lemma; Chebyshev space; periodic functions
PDFBibTeX
XMLCite

\textit{G. Thorbergsson} and \textit{M. Umehara}, Nagoya Math. J. 173, 85--138 (2004; Zbl 1066.51007)

### References:

[1] | Nagoya Math. J. 167 pp 55– (2002) · Zbl 1088.53049 · doi:10.1017/S0027763000025435 |

[2] | Amer. Math. Soc., Providence, R.I. pp 185– (1999) |

[3] | Enseign. Math. (2) 43 pp 3– (1997) |

[4] | DOI: 10.1007/BF01587937 · Zbl 0972.58003 · doi:10.1007/BF01587937 |

[5] | Geometriae Dedicata 31 pp 137– (1989) |

[6] | (1967) |

[7] | C. R. Séance Soc. Math. France, année pp 41– (1933) |

[8] | J. Reine Angew. Math. 240 pp 339– (1969) |

[9] | Bull. Calcutta Math. Soc. 1 pp 31– (1909) |

[10] | The Arnold-Gelfand Mathematical Seminars, Birkhäuser, Boston pp 257– (1997) · Zbl 0857.00029 |

[11] | Christiaan Huygens, 2 (1922/23) pp 315– |

[12] | Disconjugacy, Lecture Notes in Math. 220 (1971) |

[13] | DOI: 10.1090/S0002-9904-1944-08190-1 · Zbl 0060.34909 · doi:10.1090/S0002-9904-1944-08190-1 |

[14] | DOI: 10.1007/BF01186545 · Zbl 0003.40903 · doi:10.1007/BF01186545 |

[15] | DOI: 10.1007/BF03014758 · doi:10.1007/BF03014758 |

[16] | Band 7 (1923) |

[17] | Abh. Math. Sem. Univ. Hamburg 20 pp 196– (1956) · Zbl 0071.15601 · doi:10.1007/BF03374558 |

[18] | Birkhäuser, Boston pp 11– (1996) |

[19] | Equations différentielles ordinaires, Editons Mir (1974) |

[20] | Amer. Math. Soc., Providence, R.I. pp 229– (1999) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.