  We provide general estimates which compare the quermassintegrals of a convex
body $K$ in ${\mathbb R}^n$ with the averages of the corresponding
quermassintegrals of the $k$-codimensional sections of $K$ over $G_{n,n-k}$. An
example is the inequality
$$\alpha_{n,k,j}\frac{W_j(K)}{|K|}\leq\int_{G_{n,n-k}}\frac{W_j(K\cap
F)}{|K\cap F|}d\nu_{n,n-k}(F)\leq \beta_{n,k,j}\frac{W_j(K)}{|K|}$$ where the
constants $\alpha_{n,k,j}$ and $\beta_{n,k,j}$ depend only on $n,k$ and $j$,
which holds true for any centrally symmetric convex body $K$ in ${\mathbb R}^n$
and any $0\leq j\leq n-k-1\leq n-1$. Using these estimates we obtain some
positive results for suitable versions of the slicing problem for the
quermassintegrals of a convex body.
