Diameters of a class of smooth functions in the space $L_2$

The class $V_\psi$, consisting of the smooth functions $f(t)$, $0\le t\le1$, satisfying the condition $\int_0^1\psi[f^{(r)}(t)]\,dt\le1$, where the function $\psi(t)$ is nonnegative and $r$ is a natural number, is studied. Under certain restrictions on the function $\psi(t)$ ensuring the compactness of the class $V_\psi$, the order of decrease of the Kolmogorov diameters $d_n(V_\psi)$ is computed. The analogous problem for the case $r=1$ is solved also for functions of several variables.