论文标题
均匀的下限，符合球形多面体超平面投影的规范
A uniform lower bound on the norms of hyperplane projections of spherical polytopes
论文作者
论文摘要
令$ k $为中央对称的球形和简单的polytope，其顶点形成$ \ frac {1} {4n} - $ $ nut unit sphere in $ \ mathbb {r}^n $。我们证明了所有超平面投影的规范$ p：x \ to x $的统一下限，其中$ x $是$ n $二维规范空间，带有单位球$ k $。该估算是根据顶点的决定因素和$ k $的决定因素的。特别是，如果$ n \ geq n^{4n} $和$ k = \ cons \ {\ pm x_1，\ pm x_2，\ ldots，\ pm x_n \} $，其中$ x_1，x_2，x_2，x_2，\ ldots，x_n $是独立的随机点，则在每个单位$ pphore $ pp nitibles $ pphore $ pphore $ pphore $ p.不平等$ \ | p \ | _x \ geq 1+c_nn^{ - （2n^2+4n+6）} $（对于某些明确的常数$ C_N $），概率至少$ 1- \ frac {3} {n} {n} {n} {n}。$
Let $K$ be a centrally symmetric spherical and simplicial polytope, whose vertices form a $\frac{1}{4n}-$net in the unit sphere in $\mathbb{R}^n$. We prove a uniform lower bound on the norms of all hyperplane projections $P: X \to X$, where $X$ is the $n$-dimensional normed space with the unit ball $K$. The estimate is given in terms of the determinant function of vertices and faces of $K$. In particular, if $N \geq n^{4n}$ and $K = \conv \{ \pm x_1, \pm x_2, \ldots, \pm x_N \}$, where $x_1, x_2, \ldots, x_N$ are independent random points distributed uniformly in the unit sphere, then every hyperplane projection $P: X \to X$ satisfies an inequality $\|P\|_X \geq 1+c_nN^{-(2n^2+4n+6)}$ (for some explicit constant $c_n$), with the probability at least $1 - \frac{3}{N}.$
