Upper bounds for the volume and diameter of $m$-dimensional sections of convex bodies

In this paper some upper bounds for the volume and diameter of central sections of symmetric convex bodies are obtained in terms of the isotropy constant of the polar body. The main consequence is that every symmetric convex body $K$ in $\mathbb{R}^n$ of volume one has a proportional section $K\cap F$, dim$F= \lambda n$ ( $0< \lambda < 1$), of diameter bounded by

$R(K\cap F)\leq \frac {Cn^{3/4}\log (n+1)}{(1-\lambda)^3L_{K^\circ}}\, ,$

whenever the polar body $K^\circ$ is in isotropic position ($C>0$ is some absolute constant).