A study on various generalizations of Generalized centers in Banach spaces

In [{\em Generalized centers of finite sets in Banach spaces}, Acta Math. Univ. Comenian. (N.S.) {\bf 66}(1) (1997), 83–115], Veselý developed the idea of generalized centers for finite sets in Banach spaces. In this work, we explore the concept of {\it restricted $\text{mr}F$-center property} for a triplet $(X,Y,\text{mc}F(X))$, where $Y$ is a subspace of a Banach space $X$ and $\text{mc}F(X)$ is the family of finite subsets of $X$. In addition, we generalize the analysis to include all closed, bounded subsets of $X$. Similar to how Lindenstrauss characterized $n\mathrm{.2.}I\mathrm{.}P\mathrm{.}$, we characterize $n\mathrm{.}X\mathrm{.}I\mathrm{.}P\mathrm{.}$. So, it is possible to figure out that $Y$ has $n\mathrm{.}X\mathrm{.}I\mathrm{.}P\mathrm{.}$ in $X$ for all natural numbers $n$ if and only if $\text{tr}{rad}_Y(F)=\text{tr}{rad}_X(F)$ for all finite subsets $F$ of $Y$. It then turns out that, for all continuous, monotone functions $f$, the $f$-radii viz. $\text{tr}{rad}_Y^f(F),\text{tr}{rad}_X^f(F)$ are same whenever the generalized radii viz. $\text{tr}{rad}_Y(F),\text{tr}{rad}_X(F)$ are also same, for all finite subsets $F$ of $Y$. We establish a variety of characterizations of central subspaces of Banach spaces. With reference to an appropriate subfamily of closed and bounded subsets, it appears that a number of function spaces and subspaces exhibit the restricted weighted Chebyshev center property.