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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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