Inequalities for the quermassintegrals of sections of convex bodies

Abstract

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 $${\alpha }_{n,k,j}\frac{W_j(K)}{|K|}\le {\int }_{G_{n,n-k}}\frac{W_j(K\cap F)}{|K\cap F|}d{\nu }_{n,n-k}(F)\le {\beta }_{n,k,j}\frac{W_j(K)}{|K|}$$ 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 $0\le j\le n-k-1\le n-1$. Using these estimates we obtain some positive results for suitable versions of the slicing problem for the quermassintegrals of a convex body.