The purpose of this note is to characterize Attouch-Wets convergence for sequences of proper lower semicontinuous convex functions defined on a Banach space X in terms of the behavior of an operator
Δ
\Delta
defined on the space of such functions with values in
X
×
R
×
X
∗
X \times R \times {X^ \ast }
, defined by
Δ
(
f
)
=
{
(
x
,
f
(
x
)
,
y
)
:
(
x
,
y
)
∈
∂
f
}
\Delta (f) = \{ (x,f(x),y):(x,y) \in \partial f\}
. We show that
⟨
f
n
⟩
\langle {f_n}\rangle
is Attouch-Wets convergent to f if and only if points of
Δ
(
f
)
\Delta (f)
lying in a fixed bounded set can be uniformly approximated by points of
Δ
(
f
n
)
\Delta ({f_n})
for large n. The operator
Δ
\Delta
is a natural carrier of the Borwein variational principle, which is a key tool in both directions of our proof.