Estimates for measures of lower dimensional sections of convex bodies

TL;DR

This paper extends existing results on measures of sections of convex bodies to non-symmetric cases using a new approach involving the Blaschke-Petkantschin formula and dual affine quermassintegrals.

Contribution

It introduces an alternative method to analyze measures of convex body sections, extending results to non-symmetric convex bodies and measures with non-negative densities.

Findings

Derived bounds for measures of convex body sections in non-symmetric cases.

Extended Koldobsky's results to measures with non-negative densities.

Provided new estimates involving dual affine quermassintegrals.

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.