On stability of restricted center properties and continuity of restricted center map under $\ell_p$-direct sum

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.