On stability of restricted center properties and continuity of restricted center map under $\ell_p$-direct sum
Gayathri P, V Thota
TL;DR
The paper investigates how restricted center properties and the restricted center map behave under $\ell_p$-direct sums with $1 \le p < \infty$. It develops a radius-decomposition framework for $\ell_p$ sums, connects $P_2$ to quasi uniform rotundity and uniform Hausdorff continuity, and introduces property-$(lP_2)$ as a practical, intermediate condition ensuring the continuity of the restricted center map. Key findings include that the restricted center property, property-$(P_1)$, and semi-continuity of the restricted center map are preserved under $\ell_p$-sums; property-$(P_2)$ remains stable under finite sums but not under infinite sums; and property-$(lP_2)$ provides a stable, sufficient condition for continuity, with corresponding equivalences between componentwise and sum formulations. The results illuminate the relationship between proximinality, rotundity in subspaces, and continuity of center maps, with implications for well-posedness and approximation in product Banach spaces, especially in finite dimensions where certain coincidences occur.
Abstract
We study the stability of various restricted center properties and certain continuity properties of the restricted center map. We observe that restricted center property, property-$(P_1)$ and semi-continuity properties of the restricted center map are preserved under $\ell_p$-direct sum $(1 \leq p < \infty).$ It is shown that property-$(P_2)$ is stable under finite $\ell_p$-direct sum, but not under infinite $\ell_p$-direct sum. Additionally, we introduce a notion called property-$(lP_2)$ as a sufficient condition for the continuity of the restricted center map. Further, the stability of property-$(lP_2)$ is established.