We discuss a set-valued generalization of strong proximinality in Banach spaces, introduced by J. Mach [Continuity properties of Chebyshev centers, J. Approx. Theory 29/3 (1980) 223–230] as property-
(P_1)
(
P
1
)
. For a Banach space
X
X
, a closed convex subset
V
V
of
X
X
and a subclass
\mathcal{F}
F
of the closed bounded subsets of
X
X
, this property, defined for the triplet
(X,V,\mathcal{F})
(
X
,
V
,
F
)
, describes simultaneous strong proximinality of
V
V
at each of the sets in
\mathcal{F}
F
. We establish that if the closed unit ball of a closed subspace of a Banach space
X
X
possesses property-
(P_1)
(
P
1
)
for each of the classes of closed bounded, compact and finite subsets of
X
X
, then so does the subspace. It is also proved that the closed unit ball of an
M
M
-ideal in an
L_{1}
L
1
-predual space satisfies property-
(P_{1})
(
P
1
)
for the compact subsets of the space. For a Choquet simplex
K
K
, we provide a sufficient condition for the closed unit ball of a finite co-dimensional closed subspace of
A(K)
A
(
K
)
to satisfy property-
(P_{1})
(
P
1
)
for the compact subsets of
A(K)
A
(
K
)
. This condition also helps to establish the equivalence of strong proximinality of the closed unit ball of a finite co-dimensional subspace of
A(K)
A
(
K
)
and property-
(P_1)
(
P
1
)
of the closed unit ball of the subspace for the compact subsets of
A(K)
A
(
K
)
. Further, for a compact Hausdorff space
S
S
, a characterization is provided for a strongly proximinal finite co-dimensional closed subspace of
C(S)
C
(
S
)
in terms of property-
(P_{1})
(
P
1
)
of the subspace and that of its closed unit ball for the compact subsets of
C(S)
C
(
S
)
. We generalize this characterization for a strongly proximinal finite co-dimensional closed subspace of an
L_{1}
L
1
-predual space. As a consequence, we prove that such a subspace is a finite intersection of hyperplanes such that the closed unit ball of each of these hyperplanes satisfy property-
(P_1)
(
P
1
)
for the compact subsets of the
L_1
L
1
-predual space and vice versa. We conclude this article by providing an example of a closed subspace of a non-reflexive Banach space which satisfies
1 \frac{1}{2}
1
1
2
-ball property and does not admit restricted Chebyshev center for a closed bounded subset of the Banach space.