Estimates for measures of lower dimensional sections of convex bodies

We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if $K$ is a convex body in ${\mathbb R}^n$ with $0\in {\rm int}(K)$ and if $\mu$ is a measure on ${\mathbb R}^n$ with a locally integrable non-negative density $g$ on ${\mathbb R}^n$, then \begin{equation*}\mu (K)\leq \left (c\sqrt{n-k}\right )^k\max_{F\in G_{n,n-k}}\mu (K\cap F)\cdot |K|^{\frac{k}{n}}\end{equation*} for every $1\leq k\leq n-1$. Also, if $\mu$ is even and log-concave, and if $K$ is a symmetric convex body in ${\mathbb R}^n$ and $D$ is a compact subset of ${\mathbb R}^n$ such that $\mu (K\cap F)\leq \mu (D\cap F)$ for all $F\in G_{n,n-k}$, then \begin{equation*}\mu (K)\leq \left (ckL_{n-k}\right )^{k}\mu (D),\end{equation*} where $L_s$ is the maximal isotropic constant of a convex body in ${\mathbb R}^s$. Our method employs a generalized Blaschke-Petkantschin formula and estimates for the dual affine quermassintegrals.