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We analyze smooth nonlinear mappings for Hilbert and Banach spaces that carrysmall balls to convex sets, provided that the radius of the balls is smallenough. Being focused on the study of new and mild sufficient conditions for anonlinear mapping of Hilbert and Banach spaces to be locally convex, we addressa suitably reformulated local convexity problem analyzed within theLeray-Schauder homotopy method approach for Hilbert spaces, and within theLipscitz smoothness condition both for Hilbert and Banach spaces. Some of theresults presented in the work prove to be interesting and novel even forfinite-dimensional problems. Open problems related to the local convexityproperty for nonlinear mapping of Banach spaces are also formulated.

Some remarks on smooth mappings of Hilbert and Banach spaces and their local convexity property