The lower dimensional slicing inequality for functions and related distance inequalities

TL;DR

This paper investigates the bounds of the lower dimensional slicing inequality for functions on convex bodies, providing new lower bounds that complement existing upper bounds, and explores related distance inequalities in high-dimensional geometry.

Contribution

It constructs a convex body and density function demonstrating a lower bound for the outer volume ratio distance, complementing known upper bounds and refining understanding of slicing inequalities.

Findings

Established a lower bound for the outer volume ratio distance for convex bodies.

Provided a matching order of magnitude for the bounds in high dimensions.

Extended previous results for the case k=1 to general k.

Abstract

It was shown in [11] that for every origin-symmetric star body $K \subseteq \mathbb R^n$ of volume $1$, every even continuous probability density $f$ on $K$ and $1 \leq k \leq n-1$, there exists a subspace $F \subseteq \mathbb R^n$ of codimension $k$ such that \[ \int_{K \cap F} f \geq c^k (d_{\rm ovr}(K, \mathcal{BP}_k^n))^{-k} \] where $d_{\rm ovr}(K, \mathcal{BP}_k^n)$ is the outer volume ratio distance from $K$ to the class of generalized $k$-intersection bodies, and $c>0$ is a universal constant. The upper bound $d_{\rm ovr}(K, \mathcal{BP}_k^n) \leq c' \sqrt{n/k} \left(\log\left(\frac{en}k\right)\right)^{3/2}$ was established in [13] for every origin-symmetric convex body $K$. In this note we show that there exist an origin-symmetric convex body $K$ of volume $1$ and an even continuous probability density $f$ supported on $K$ such that for every subspace $F$ of codimension $k$, \[ \int_{K \cap F} f \leq \left( c \sqrt{\frac n{k \log(n)} } \right)^{-k}. \] As a consequence we obtain a lower bound for $d_{\rm ovr}(K, \mathcal{BP}_k^n)$ with $K$ a convex body, complementing the upper bound in \cite{koldobsky2011isomorphic}. This is \[c \sqrt{n/k} (\log(n))^{-1/2} \leq \sup_K d_{\rm ovr}(K, \mathcal{BP}_k^n) \leq c' \sqrt{n/k} \left(\log\left(\frac{en}k\right)\right)^{3/2}.\] The case $k=1$ was obtained previously in [5,6].