On local convexity of nonlinear mappings between Banach spaces

Abstract

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 e{open} < c the image f(B e{open}(x)) of each e{open}-ball B e{open}(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip 2(f), the Lipschitz-open constant Lip o(f) of f, and the 2-convexity number conv 2(X) of the Banach space X. © 2012 Versita Warsaw and Springer-Verlag Wien.