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

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