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Start Over Creator Duda, Jakub

1. Curves in Banach spaces which allow a $C^{1,\rm BV}$ parametrization or a parametrization with finite convexity

Creator:
Duda, Jakub and Zajíček, Luděk
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
curve in Banach spaces, $C^{1,\rm BV}$ parametrization, and parametrization with bounded convexity
Language:
English
Description:
We give a complete characterization of those $f\colon [0,1] \to X$ (where $X$ is a Banach space) which allow an equivalent $C^{1,\rm BV}$ parametrization (i.e., a $C^1$ parametrization whose derivative has bounded variation) or a parametrization with bounded convexity. Our results are new also for $X= \mathbb R^n$. We present examples which show applicability of our characterizations. For example, we show that the $C^{1,\rm BV}$ and $C^2$ parametrization problems are equivalent for $X=\mathbb R$ but are not equivalent for $X = \mathbb R^2$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. Curves with finite turn

Creator:
Duda, Jakub
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
curve with finite turn, tangent of a curve, curve with finite convexity, delta-convex curve, and d.c. curve
Language:
English
Description:
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.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public