论文标题
关于凸形的投影的体积比
On the volume ratio of projections of convex bodies
论文作者
论文摘要
我们研究了两个凸体的投影之间的体积比。鉴于高维凸面$ k $，我们表明还有另一个凸件$ l $，使得固定级别$ k $和$ l $的任何两个固定排名之间的体积比很大。也就是说，我们证明，每$ 1 \ leq k \ leq n $，对于每个凸件$ k \ subset \ mathbb {r}^n $，都有一个集中的对称体$ l \ subset \ subset \ subset \ subset \ mathbb {r}^n $排名$ k $一个有$$ \ mbox {vr}（pk，ql）\ geq c \，\ min \ left \ {\ frac {\ frac {\ sqrt {\ sqrt {n}} \，\ sqrt {\ sqrt {\ frac {1} \，， \ frac {\ sqrt {k}}} {\ sqrt {\ log（\ frac {n \ log（n）} {k}）}}}}}} \ right \}，$ $ 其中$ c> 0 $是绝对常数。该总体下限在制度$ k \ geq n^{2/3} $中是锋利的（达到对数因素）。
We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body $K$ we show that there is another convex body $L$ such that the volume ratio between any two projections of fixed rank of the bodies $K$ and $L$ is large. Namely, we prove that for every $1\leq k\leq n$ and for each convex body $K\subset \mathbb{R}^n$ there is a centrally symmetric body $L \subset \mathbb{R}^n$ such that for any two projections $P, Q: \mathbb{R}^n \to \mathbb{R}^n$ of rank $k$ one has $$ \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 $k\geq n^{2/3}$.
