On the volume ratio of projections of convex bodies

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 $1\le k\le n$ 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 $$vr(PK,QL)\ge c\min \{\frac{k}{\sqrt{n}}\sqrt{\frac{1}{\log \log \log (\frac{n\log (n)}{k})}},\frac{\sqrt{k}}{\sqrt{\log (\frac{n\log (n)}{k})}}\},$$ where $c>0$ is an absolute constant. This general lower bound is sharp (up to logarithmic factors) in the regime $k\ge n^{2\mathrm{/}3}$.