  We study volumes of sections of convex origin-symmetric bodies in
$\mathbb{R}% ^{n}$ induced by orthonormal systems on probability spaces. The
approach is based on volume estimates of \ John-L\"{o}wner ellipsoids and
expectations of norms induced by the respective systems. The estimates obtained
allows us to establish lower bounds for the radii of sections which gives lower
bounds for Gelfand widths (or linear cowidths). As an application we offer a
new method of evaluation of Gelfand and Kolmogorov widths of multiplier
operators. In particular, we establish sharp orders of widths of standard
Sobolev classes $W_{p}^{\gamma }$ in $L_{q}$ in the difficult case, i.e. $%
1<q\leq p\leq \infty $.
