  Let $P(D)$ be the differential operator induced by a polynomial $P$, and let
${U^{[P]}_2}$ be the class of multivariate periodic functions $f$ such that
$\|P(D)(f)\|_2\leq 1$. The problem of computing the asymptotic order of the
Kolmogorov $n$-width $d_n({U^{[P]}_2},L_2)$ in the general case when
${U^{[P]}_2}$ is compactly embedded into $L_2$ has been open for a long time.
In the present paper, we use convex analytical tools to solve it in the case
when $P(D)$ is non-degenerate.
