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arXiv · 1205.3412

On Local Convexity Of Nonlinear Mappings Between Banach Spaces

Iryna Banakh·Taras Banakh·Anatolij Plichko·Anatoliy Prykarpatsky
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Abstract

We find conditions for a smooth nonlinear map $f:U\rightarrow V$ between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some $c$ and each positive $\varepsilon<c$ the image $% f(B_\varepsilon(x))$ of each $\varepsilon$-ball $B_\varepsilon(x)\subset U$ is convex. We give a lower bound on $c$ via the second order Lipschitz constant $\mathrm{Lip}_2(f)$, the Lipschitz-open constant $\mathrm{Lip}_o(f)$ of $f$, and the 2-convexity number $\mathrm{conv}_2(X)$ of the Banach space $% X$.

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Iryna Banakh, Taras Banakh, Anatolij Plichko, Anatoliy Prykarpatsky. 2012-05-12. On Local Convexity Of Nonlinear Mappings Between Banach Spaces. https://arxiv.org/abs/1205.3412

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