On Local Convexity Of Nonlinear Mappings Between Banach Spaces

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 $$ of each $\epsilon$-ball $B_{\epsilon }(x)\subset U$ is convex. We give a lower bound on $c$ via the second order Lipschitz constant ${\mathrm{L}\mathrm{i}\mathrm{p}}_2(f)$, the Lipschitz-open constant ${\mathrm{L}\mathrm{i}\mathrm{p}}_o(f)$ of $f$, and the 2-convexity number ${\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}_2(X)$ of the Banach space $$.