Inequalities for the Gamma function and estimates for the volume of sections of $B^n_p$

Let $B^n_p=\{(x_i)\in\mathbb{R}^n;\sum_1^n\vert x_i\vert^p\leq1\}$ and let $E$ be a $k$-dimensional subspace of $\mathbb{R}^n$. We prove that $\vert E\cap B^n_p\vert _k^{1/k}\geq \vert B^n_p\vert _n^{1/n}$, for $1\leq k\leq (n-1)/2$and $k=n-1$ whenever $1<p<2$. We also consider $0<p<1$ and other related cases. We obtain sharp inequalities involving Gamma function in order to get these results.