Near equality, near convexity, sums of maximally monotone operators, and averages of firmly nonexpansive mappings (Q353136)

In this paper, the authors study the notions of near equality and near convexity in the context of real Euclidean spaces. Let \(X\) be a real Euclidean space. Two subsets \(A\) and \(B\) of \(X\) are called \textit{nearly equal} if \(\overline{A}=\overline{B}\) and \(\operatorname{ri} A=\operatorname{ri}B\). This is denoted as \(A\approx B\). In addition, a subset \(A\) is called \textit{nearly convex} if there exists a convex subset \(C\) such that \(C\subseteq A\subseteq \overline{C}\). Thus the first main result of this paper states that, given a family \((A_i)_{i\in I}\) of nearly convex subsets and a family \((B_i)_{i\in I}\) such that \(A_i\approx B_i\) for all \(i\), then \(\sum_{i\in I}A_i\) is nearly convex and \[ \sum_{i\in I}A_i \approx \sum_{i\in I}B_i . \] Furthermore, the authors apply this to the study of the ranges of a certain kind of monotone operators and then to firmly non-expansive mappings. Namely, the range of an average of firmly non-expansive mappings is nearly equal to the average of the ranges of such mappings. Finally, they prove that the average of asymptotically regular mappings is asymptotically regular.