Displaying similar documents to “On vector functions of bounded convexity”

On inverses of δ -convex mappings

Jakub Duda (2001)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In the first part of this paper, we prove that in a sense the class of bi-Lipschitz δ -convex mappings, whose inverses are locally δ -convex, is stable under finite-dimensional δ -convex perturbations. In the second part, we construct two δ -convex mappings from ℓ 1 onto ℓ 1 , which are both bi-Lipschitz and their inverses are nowhere locally δ -convex. The second mapping, whose construction is more complicated, has an invertible strict derivative at 0 . These mappings show that for (locally) δ -convex...

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

Similarity:

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 = ℝ 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 = ℝ but are not equivalent for X = ℝ 2 .