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

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 = a"e (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 = a"e but are not equivalent for X = a"e (n) .