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 (4n)^{-1}-net in the unit sphere in {mathbb R}^n. We prove a uniform lower bound on the norms of all hyperplane projections P:Xrightarrow 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 Nge n^{4n} and K={{,textrm{conv},}}{{pm x_1,pm x_2,dots ,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 rightarrow X satisfies an inequality Vert PVert _Xge 1+c_nN^{-(2n^2+4n+6)} (for some explicit constant c_n), with the probability at least 1-3/N.