2021/10/28 by Yarema A. Prykarpatskyy, Prykarpatskyy, Yarema A., Pukach, Petro Ya. +2
Computer Science · Mathematics · #46B20 #49J50 #52A41 #58C20 #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2110.15175
openalex publication_date 2021/10/28 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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.