Reconstruction of functions with the given modulus of continuity

Let $\mu$ be a measure on $[a,b],C_{\omega,\mu}$ the class of all functions $f$ whose continuity modulus does not exceed the given function $\omega$ and such that $\int_a^bf\,d\mu=0$. The problem of eatimatinq $\operatorname{diam}C_{\omega,\mu}$ (in the space $C([a,b])$) is reduced to an equilibrium problem for the “potential” $\int\omega(|x-t|)\,d\mu(t)$.