On Local Convexity Of Nonlinear Mappings Between Banach Spaces

Abstract

We find conditions for a smooth nonlinear map $f:U\to V$ between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some $c$ and each positive $\epsilon <c$ the image of each $\epsilon$-ball $B_{\epsilon }(x)\subset U$ 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 .

6 pages