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By using D. Preiss' approach to a construction from a paper by J. Matoušek and E. Matoušková, and some results of E. Matoušková, we prove that we can decompose a separable Banach space with modulus of convexity of power type p as a union of a ball small set (in a rather strong symmetric sense) and a set which is Aronszajn null. This improves an earlier unpublished result of E. Matoušková. As a corollary, in each separable Banach space with modulus of convexity of power type p, there exists a closed...

We find conditions on a real function f:[a,b] → ℝ equivalent to being Lebesgue equivalent to an n-times differentiable function (n ≥ 2); a simple solution in the case n = 2 appeared in an earlier paper. For that purpose, we introduce the notions of $CBV{G}_{1/n}$ and $SBV{G}_{1/n}$ functions, which play analogous rôles for the nth order differentiability to the classical notion of a VBG⁎ function for the first order differentiability, and the classes $CB{V}_{1/n}$ and $SB{V}_{1/n}$ (introduced by Preiss and Laczkovich) for Cⁿ smoothness. As a consequence,...

In the first part of this paper, we prove that in a sense the class of bi-Lipschitz $\delta$-convex mappings, whose inverses are locally $\delta$-convex, is stable under finite-dimensional $\delta$-convex perturbations. In the second part, we construct two $\delta$-convex mappings from ${\ell }_1$ onto ${\ell }_1$, which are both bi-Lipschitz and their inverses are nowhere locally $\delta$-convex. The second mapping, whose construction is more complicated, has an invertible strict derivative at $0$. These mappings show that for (locally) $\delta$-convex mappings...

In this paper we study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by Alexandrov, Pogorelov, and Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces. One of the implications was only proved in the finite dimensional context previously, and equivalence of finiteness...

We give a complete characterization of those $f:[0,1]\to X$ (where $X$ is a Banach space) which allow an equivalent $C^{1,\mathrm{BV}}$ 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 $C^{1,\mathrm{BV}}$ and $C^2$ parametrization problems are equivalent for $X=ℝ$ but are not equivalent for $X={ℝ}^2$.