  We analyze smooth nonlinear mappings for Hilbert and Banach spaces that carry
small balls to convex sets, provided that the radius of the balls is small
enough. Being focused on the study of new and mild sufficient conditions for a
nonlinear mapping of Hilbert and Banach spaces to be locally convex, we address
a suitably reformulated local convexity problem analyzed within the
Leray-Schauder homotopy method approach for Hilbert spaces, and within the
Lipscitz smoothness condition both for Hilbert and Banach spaces. Some of the
results presented in the work prove to be interesting and novel even for
finite-dimensional problems. Open problems related to the local convexity
property for nonlinear mapping of Banach spaces are also formulated.
