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

TL;DR

This paper investigates conditions under which smooth nonlinear mappings in Hilbert and Banach spaces transform small balls into convex sets, introducing new criteria and reformulating the local convexity problem using homotopy and Lipschitz smoothness.

Contribution

It provides new mild sufficient conditions for local convexity of nonlinear mappings in Hilbert and Banach spaces, including reformulations using homotopy and Lipschitz smoothness.

Findings

New sufficient conditions for local convexity in Hilbert and Banach spaces.

Reformulation of the local convexity problem using Leray-Schauder homotopy.

Results are novel even for finite-dimensional cases.

Abstract

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.