Finite-dimensional normed spaces without small-norm hyperplane projections

We prove that there exists an absolute constant $c>1$ such that for every $n \geq 3$ there exists an $n$-dimensional real normed space $X$ for which every linear projection $P$ onto a hyperplane satisfies $\|P\| \geq c$. More generally, for every integer $k \geq 1$ there exists a constant $c_k>1$ su