Curves with finite turn

Jakub Duda

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 1, page 23-49
  • ISSN: 0011-4642

Abstract

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In this paper 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 Alexandrov, Pogorelov, and 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 (for example Corollary 5.12 concerning the permanent properties of curves with finite turn) than those that were proved previously with geometric methods in Euclidean spaces.

How to cite

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Duda, Jakub. "Curves with finite turn." Czechoslovak Mathematical Journal 58.1 (2008): 23-49. <http://eudml.org/doc/31197>.

@article{Duda2008,
abstract = {In this paper 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 Alexandrov, Pogorelov, and 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 (for example Corollary 5.12 concerning the permanent properties of curves with finite turn) than those that were proved previously with geometric methods in Euclidean spaces.},
author = {Duda, Jakub},
journal = {Czechoslovak Mathematical Journal},
keywords = {curve with finite turn; tangent of a curve; curve with finite convexity; delta-convex curve; d.c. curve; tangent of a curve; curve with finite convexity; delta-convex curve},
language = {eng},
number = {1},
pages = {23-49},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Curves with finite turn},
url = {http://eudml.org/doc/31197},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Duda, Jakub
TI - Curves with finite turn
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 1
SP - 23
EP - 49
AB - In this paper 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 Alexandrov, Pogorelov, and 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 (for example Corollary 5.12 concerning the permanent properties of curves with finite turn) than those that were proved previously with geometric methods in Euclidean spaces.
LA - eng
KW - curve with finite turn; tangent of a curve; curve with finite convexity; delta-convex curve; d.c. curve; tangent of a curve; curve with finite convexity; delta-convex curve
UR - http://eudml.org/doc/31197
ER -

References

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  1. General Theory of Irregular Curves, Mathematics and its Applications (Soviet Series), Vol.  29, Kluwer Academic Publishers, Dordrecht, 1989. (1989) MR1117220
  2. Geometric Nonlinear Functional Analysis, Vol. 1. Colloquium Publications 48, American Mathematical Society, Providence, 2000. (2000) MR1727673
  3. On D.C.  functions and mappings, Atti Sem. Mat. Fis. Univ. Modena 51 (2003), 111–138. (2003) MR1993883
  4. Konvexita a ohyb křivky, Master Thesis, Charles University, Prague, 1987. (Czech) (1987) 
  5. private communication, . 
  6. Extrinsic geometry of convex surfaces, Translations of Mathematical Monographs, Vol.  35, American Mathematical Society, Providence, 1973. (1973) Zbl0311.53067MR0346714
  7. Convex functions, Pure and Applied Mathematics, Vol. 57, Academic Press, New York-London, 1973. (1973) MR0442824
  8. Geometry of Spheres in Normed Spaces, Lecture Notes in Pure and Applied Mathematics Vol. 20, Marcel Dekker, New York-Basel, 1976. (1976) MR0467256
  9. On the multiplicity points of monotone operators on separable Banach spaces, Comment. Math. Univ. Carolinae 27 (1986), 551–570. (1986) MR0873628
  10. Delta-convex mappings between Banach spaces and applications, Diss. Math. Vol.  289, 1989. (1989) MR1016045

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