On the volume ratio of projections of convex bodies

TL;DR

This paper investigates the volume ratio between projections of convex bodies in high-dimensional spaces, establishing a lower bound that is nearly optimal for certain projection ranks.

Contribution

It proves the existence of a convex body L such that the volume ratio between any two rank-k projections of K and L is significantly large, providing a sharp lower bound.

Findings

Established a universal lower bound for volume ratios of projections.

The bound is nearly optimal for projection ranks k ≥ n^{2/3}.

Demonstrated the existence of a convex body L with specific projection properties.

Abstract

We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body $K$ we show that there is another convex body $L$ such that the volume ratio between any two projections of fixed rank of the bodies $K$ and $L$ is large. Namely, we prove that for every $1\leq k\leq n$ and for each convex body $K\subset \mathbb{R}^n$ there is a centrally symmetric body $L \subset \mathbb{R}^n$ such that for any two projections $P, Q: \mathbb{R}^n \to \mathbb{R}^n$ of rank $k$ one has $$ \mbox{vr}(PK, QL) \geq c \, \min\left\{\frac{ k}{ \sqrt{n}} \, \sqrt{\frac{1}{\log \log \log(\frac{n\log(n)}{k})}}, \, \frac{\sqrt{k}}{\sqrt{\log(\frac{n\log(n)}{k})}}\right\}, $$ where $c>0$ is an absolute constant. This general lower bound is sharp (up to logarithmic factors) in the regime $k\geq n^{2/3}$.