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Czechoslovak Mathematical Journal, Vol. 58, No. 1, pp. 23-49, 2008
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Curves with finite turn

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Jakub Duda

* Jakub Duda*, Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel, e-mail: ` jakub.duda@weizmann.ac.il`

**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.

**Keywords:** curve with finite turn, tangent of a curve, curve with finite convexity, delta-convex curve, d.c. curve

**Classification (MSC 2000):** 14H50, 46T20

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