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 $N\ge n^{4n}$ and $K = \conv \{ \pm x_1, \pm x_2, \ldots, \pm x_N \}$, where $x_1,x_2,\dots ,x_N$ are independent random points distributed uniformly in the unit sphere, then every hyperplane projection $P:X\to X$ satisfies an inequality $\Vert P{\Vert }_X\ge 1+c_n{N}^{-(2n^2+4n+6)}$ (for some explicit constant $c_n$), with the probability at least $1-\frac{3}{N}.$

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