  In [{\em Generalized centers of finite sets in Banach spaces}, Acta Math.
Univ. Comenian. (N.S.) {\bf 66}(1) (1997), 83--115], Vesel\'{y} developed the
idea of generalized centers for finite sets in Banach spaces. In this work, we
explore the concept of {\it restricted $\mr{F}$-center property} for a triplet
$(X,Y,\mc{F}(X))$, where $Y$ is a subspace of a Banach space $X$ and
$\mc{F}(X)$ 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 $n.2.I.P.$, we characterize $n.X.I.P.$. So, it is
possible to figure out that $Y$ has $n.X.I.P.$ in $X$ for all natural numbers
$n$ if and only if $\tr{rad}_Y(F)=\tr{rad}_X(F)$ for all finite subsets $F$ of
$Y$. It then turns out that, for all continuous, monotone functions $f$, the
$f$-radii viz. $\tr{rad}_Y^f(F),\tr{rad}_X^f(F)$ are same whenever the
generalized radii viz. $\tr{rad}_Y(F), \tr{rad}_X(F)$ 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.
