A uniform lower bound on the norms of hyperplane projections of spherical polytopes

Abstract

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 $1 - \frac{3}{N}.$