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Keywords:
curve in Banach spaces; $C^{1,\rm BV}$ parametrization; parametrization with bounded convexity
curve in Banach spaces; $C^{1,\rm BV}$ parametrization; parametrization with bounded convexity
Summary:
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 $X = \mathbb R^2$.
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