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 $0in {rm int}(K)$ 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 (csqrt{n-k}right )^kmax_{Fin G_{n,n-k}}mu (Kcap F)cdot |K|^{frac{k}{n}}end{equation*} for every $1leq kleq n-1$. 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 $mu (Kcap F)leq mu (Dcap F)$ for all $Fin G_{n,n-k}$, 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.
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