Inequalities for the quermassintegrals of sections of convex bodies

TL;DR

This paper establishes inequalities relating the quermassintegrals of convex bodies to the averages over their sections, providing new bounds and insights relevant to the slicing problem in convex geometry.

Contribution

The paper introduces general estimates comparing quermassintegrals of convex bodies with their sections, advancing understanding of geometric inequalities and the slicing problem.

Findings

Derived bounds for quermassintegrals of convex bodies.

Established inequalities involving averages over sections.

Provided positive results for the slicing problem.

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 $$\alpha_{n,k,j}\frac{W_j(K)}{|K|}\leq\int_{G_{n,n-k}}\frac{W_j(K\cap F)}{|K\cap F|}d\nu_{n,n-k}(F)\leq \beta_{n,k,j}\frac{W_j(K)}{|K|}$$ 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 $0\leq j\leq n-k-1\leq n-1$. Using these estimates we obtain some positive results for suitable versions of the slicing problem for the quermassintegrals of a convex body.