We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.
@article{bwmeta1.element.doi-10_2478_s11533-012-0101-z,
author = {Iryna Banakh and Taras Banakh and Anatolij Plichko and Anatoliy Prykarpatsky},
title = {On local convexity of nonlinear mappings between Banach spaces},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {2264-2271},
zbl = {1267.46027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0101-z}
}
Iryna Banakh; Taras Banakh; Anatolij Plichko; Anatoliy Prykarpatsky. On local convexity of nonlinear mappings between Banach spaces. Open Mathematics, Tome 10 (2012) pp. 2264-2271. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0101-z/
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