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Curves in Banach spaces which allow a $$C^{1,\text{BV}}$$ parametrization or a parametrization with finite convexity. (English) Zbl 1313.26047
Let $$X$$ be a Banach space and $$f\: [0,1] \to X$$ be a continuously parametrized curve. A general problem can be formulated as follows: For a given $$k \in \mathbb {N}$$, does there exist an equivalent parametrization of the curve which is in addition $$C^k$$-smooth. This general problem has been solved in several special cases in the literature (listed in the paper), in particular:
$$X=\mathbb {R}, C^k, k \in \mathbb {N};$$
$$X$$ arbitrary, $$C^1$$; and
$$X$$ having a Frechet smooth norm, $$C^2$$, resp. $$C^{1,\alpha }$$.
In the present paper, a characterization is obtained in the case of $$C^{1,\text{BV}}$$ smoothness (i.e., the first derivative is of bounded variation) and a parametrization with a bounded convexity, respectively. These characterizations are based on the technical concepts of turn and $$\frac {1}{2}$$-variation of the curve. It turns out that for $$X=\mathbb {R}$$ a parametrization with bounded convexity exists if and only if a $$C^2$$-smooth parametrization exists. For $$X=\mathbb {R}^2$$, an example of a curve admitting a bounded convexity parametrization which admits no $$C^1$$-smooth parametrization is provided.
##### MSC:
 26E20 Calculus of functions taking values in infinite-dimensional spaces 26A45 Functions of bounded variation, generalizations 26A51 Convexity of real functions in one variable, generalizations 53A04 Curves in Euclidean and related spaces
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