  In this paper we provide explicit upper and lower bounds on certain $L^2$
$n$-widths, i.e., best constants in $L^2$ approximation. We further describe a
numerical method to compute these $n$-widths approximately, and prove that this
method is superconvergent. Based on our numerical results we formulate a
conjecture on the asymptotic behaviour of the $n$-widths. Finally we describe
how the numerical method can be used to compute the breakpoints of the optimal
spline spaces of Melkman and Micchelli, which have recently received renewed
attention in the field of Isogeometric Analysis.
