A Uniform Lower Bound on the Norms of Hyperplane Projections of Spherical Polytopes

Abstract

AbstractLet K be a centrally symmetric spherical and simplicial polytope, whose vertices form a $$(4n)^{-1}$$ ( 4 n ) - 1 -net in the unit sphere in $${\mathbb R}^n$$ R n . We prove a uniform lower bound on the norms of all hyperplane projections $$P:X\rightarrow X$$ P : X → 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 $$N\ge n^{4n}$$ N ≥ n 4 n and $$K={{\,\textrm{conv}\,}}{\{\pm x_1,\pm x_2,\dots ,\pm x_N\}}$$ K = conv { ± x 1 , ± x 2 , ⋯ , ± x N } , where $$x_1,x_2,\dots ,x_N$$ x 1 , x 2 , ⋯ , x N are independent random points distributed uniformly in the unit sphere, then every hyperplane projection $$P:X \rightarrow X$$ P : X → X satisfies an inequality $$\Vert P\Vert _X\ge 1+c_nN^{-(2n^2+4n+6)}$$ ‖ P ‖ X ≥ 1 + c n N - ( 2 n 2 + 4 n + 6 ) (for some explicit constant $$c_n$$ c n ), with the probability at least $$1-3/N$$ 1 - 3 / N .