On Local Convexity Of Nonlinear Mappings Between Banach Spaces

TL;DR

This paper establishes conditions under which smooth nonlinear maps between Banach or Hilbert spaces are locally convex, linking geometric properties of the map and the space to convexity of images of small balls.

Contribution

It provides new criteria involving Lipschitz constants and space convexity measures for local convexity of nonlinear mappings between Banach spaces.

Findings

Derived a lower bound on the convexity radius c.

Connected local convexity to second order Lipschitz constants.

Linked convexity properties to the 2-convexity number of the space.

Abstract

We find conditions for a smooth nonlinear map $f:U\rightarrow V$ between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some $c$ and each positive $\varepsilon<c$ the image $% f(B_\varepsilon(x))$ of each $\varepsilon$-ball $B_\varepsilon(x)\subset U$ is convex. We give a lower bound on $c$ via the second order Lipschitz constant $\mathrm{Lip}_2(f)$, the Lipschitz-open constant $\mathrm{Lip}_o(f)$ of $f$, and the 2-convexity number $\mathrm{conv}_2(X)$ of the Banach space $% X$.