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Curves in Banach spaces which allow a C 1 , BV parametrization or a parametrization with finite convexity

Jakub Duda, Luděk Zajíček (2013)

Czechoslovak Mathematical Journal

We give a complete characterization of those f : [ 0 , 1 ] X (where X is a Banach space) which allow an equivalent C 1 , 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 = n . We present examples which show applicability of our characterizations. For example, we show that the C 1 , BV and C 2 parametrization problems are equivalent for X = but are not equivalent for X = 2 .

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