At first the author proves that for any logarithmically concave function \(f: {\mathbb{R}}^ k\to [0,\infty)\) with \(0<\int_{{\mathbb{R}}^ k}f<\infty\) and for any number \(p\geq 1\), the function \(\| \cdot \|: {\mathbb{R}}^ k\to {\mathbb{R}}\) defined by \[ \| x \| = \begin{cases} \left[ \int_{0}^{\infty} f(rx)r^{p-1}dr\right]^{-1/p}, & x \neq 0 \\ 0 & x=0 \end{cases} \] is a norm. By applying this result he estimates then the ratioof the volumes of sections of convex sets. More precisely, he shows that if C is an isotropic symmetric convex set in \({\mathbb{R}}^ n\), and H and K are k-codimensional subspaces of \({\mathbb{R}}^ n(k<n)\), then holds \[ | H\cap C| /| K\cap C| \leq [k(k+1)(k+2)]^{k/2}v_ k/2^ k(k!)^{1/2}< (\Pi e^ 2k)^{k/2}, \] where \(v_ k\) denotes the volume of the Euclidean ball of radius 1 in \({\mathbb{R}}^ k\).