球面多面体超平面投影范数的一致下界
A uniform lower bound on the norms of hyperplane projections of spherical polytopes
摘要：
设$K$是中心对称的球面单胞多面体，其顶点在$\mathbb{R}^n$的单位球面上构成一个$\frac{1}{4n}-$网.证明了超平面投影$P: X \to X$的范数的一致下界，其中$X$是单位球$K$的$n$维赋范空间.利用$K$的点和面的行列式函数给出了估计。特别地，如果$N \geq n^{4n}$和$K = \conv \{ \pm x_1, \pm x_2, \ldots, \pm x_N \}$，其中$x_1, x_2, \ldots, x_N$是均匀分布在单位球面中的独立随机点，则每个超平面投影$P: X \to X$满足不等式$\|P\|_X \geq 1+c_nN^{-(2n^2+4n+6)}$（对于某个显式常数$c_n$），概率至少为$1 - \frac{3}{N}.$
英文摘要：
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 $1 - \frac{3}{N}.$
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