The following results is proved: Let
A
A
be a convex bounded non weakly relatively compact subset of a Banach space
E
E
. We consider a convex weakly compact subset
D
D
of
E
E
which does not contain the origin. Then there is a sequence
\left\{x_n^*\right\}_{n\ge 1}
{
x
n
∗
}
n
≥
1
in
B_{E^*}
B
E
∗
and
g_0^*\in \hbox{co}_{\sigma}\{x_n^*:n\ge 1\}
g
0
∗
∈
co
σ
{
x
n
∗
:
n
≥
1
}
such that for all
h\in \ell_\infty (A)
h
∈
ℓ
∞
(
A
)
satisfying that for all
a\in A,
a
∈
A
,
\liminf_{n\ge 1}x_n^*(a) \le h(a) \le\limsup_{n\ge 1}x_n^*(a),
lim inf
n
≥
1
x
n
∗
(
a
)
≤
h
(
a
)
≤
lim sup
n
≥
1
x
n
∗
(
a
)
,
we have that
g_0^*- h
g
0
∗
−
h
does not attain its supremum on
A
A
and
( g_0^*- h)(d)>0
(
g
0
∗
−
h
)
(
d
)
>
0
for every
d\in D
d
∈
D
.