Let
S
N
2
S_N^2
denote the nonlinear manifold of second order splines defined on [0, 1] having at most
N
N
interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function
f
f
by elements of
S
N
2
S_N^2
. Approximation relative to the
L
2
[
0
,
1
]
{L_2}[0,1]
norm is treated first, with the results then extended to the best
L
1
{L_1}
and best one-sided
L
1
{L_1}
approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function
f
f
satisfying
f
>
0
f > 0
has a unique best approximant from
S
N
2
S_N^2
provided either
log
f
\log f
is concave, or
N
N
is sufficiently large,
N
⩾
N
0
(
f
)
N \geqslant {N_0}(f)
; for any
N
N
, there is a smooth function
f
f
, with
f
>
0
f > 0
, having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.