This is an announcement for the paper "Estimates for measures of lower dimensional sections of convex bodies" by Giorgos Chasapis, Apostolos Giannopoulos and Dimitris-Marios Liakopoulos.

Abstract: 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.

Archive classification: math.MG math.FA

Submitted from: gchasapis@math.uoa.gr

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1512.08393

or

http://arXiv.org/abs/1512.08393