Curves with finite turn.

*(English)*Zbl 1167.46321Summary: We study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by A. D. Alexandrov, A. V. Pogorelov, and Yu. Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces. One of the implications was only proved in the finite dimensional context previously, and equivalence of finiteness of turn with finiteness of turn of tangents in arbitrary Banach spaces. We also develop an auxiliary theory of one-sidedly smooth curves with values in Banach spaces. We use analytic language and methods to provide analogues of angular theorems. In some cases our approach yields stronger results than those that were proved previously with geometric methods in Euclidean spaces.

##### MSC:

46T99 | Nonlinear functional analysis |

53A04 | Curves in Euclidean and related spaces |

58B99 | Infinite-dimensional manifolds |

##### References:

[1] | A. D. Alexandrov, Yu. Reshetnyak: General Theory of Irregular Curves. Mathematics and its Applications (Soviet Series), Vol. 29. Kluwer Academic Publishers, Dordrecht, 1989. |

[2] | Y. Benyamini, J. Lindenstrauss: Geometric Nonlinear Functional Analysis, Vol. 1. Colloquium Publications 48. American Mathematical Society, Providence, 2000. · Zbl 0946.46002 |

[3] | J. Duda, L. Veselý, L. Zajíček: On D.C. functions and mappings. Atti Sem. Mat. Fis. Univ. Modena 51 (2003), 111–138. · Zbl 1072.46025 |

[4] | M. Gronychová: Konvexita a ohyb křivky. Master Thesis. Charles University, Prague, 1987. (In Czech.) |

[5] | N. Kalton: private communication. |

[6] | A. V. Pogorelov: Extrinsic geometry of convex surfaces. Translations of Mathematical Monographs, Vol. 35. American Mathematical Society, Providence, 1973. · Zbl 0311.53067 |

[7] | A. W. Roberts, D. E. Varberg: Convex functions. Pure and Applied Mathematics, Vol. 57. Academic Press, New York-London, 1973. |

[8] | J. J. Schäffer: Geometry of Spheres in Normed Spaces. Lecture Notes in Pure and Applied Mathematics Vol. 20. Marcel Dekker, New York-Basel, 1976. · Zbl 0344.46038 |

[9] | L. Veselý: On the multiplicity points of monotone operators on separable Banach spaces. Comment. Math. Univ. Carolinae 27 (1986), 551–570. · Zbl 0616.47043 |

[10] | L. Veselý, L. Zajíček: Delta-convex mappings between Banach spaces and applications. Diss. Math. Vol. 289. 1989. |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.