Let
H
H
be a finite-dimensional Hilbert space,
B
(
H
)
B\left ( H \right )
the space of bounded linear operators on
H
H
, and
C
C
a convex subset of
B
(
H
)
B\left ( H \right )
. If
A
A
is a fixed operator in
B
(
H
)
B\left ( H \right )
, then
A
A
has a unique best approximant from
C
C
in the
C
P
{C_P}
norm for
1
>
p
>
∞
1 > p > \infty
. However, in the
C
1
{C_1}
(trace) norm,
A
A
may have many best approximants from
C
C
. In this paper, it is shown that the best
C
p
{C_p}
approximants to
A
A
converge to a select trace class approximant
A
1
{A_1}
as
p
→
1
p \to 1
. Furthermore,
A
1
{A_1}
is the unique trace class approximant minimizing
∑
i
=
1
n
S
i
(
A
−
B
)
ln
S
i
(
A
−
B
)
\sum \nolimits _{i = 1}^n {{S_i}\left ( {A - B} \right )\operatorname {ln }{S_i}\left ( {A - B} \right )}
over all trace class approximants
B
B
. The numbers
S
i
(
T
)
{S_i}\left ( T \right )
are the eigenvalues of the positive part
|
T
|
\left | T \right |
of
T
T
.