# Curves with finite turn

Czechoslovak Mathematical Journal (2008)

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

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topDuda, 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 -

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## Citations in EuDML Documents

top- Libor Veselý, Luděk Zajíček, On vector functions of bounded convexity
- Jakub Duda, Luděk Zajíček, Curves in Banach spaces which allow a ${C}^{1,\mathrm{BV}}$ parametrization or a parametrization with finite convexity
- Luděk Zajíček, A note on propagation of singularities of semiconcave functions of two variables

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