Finite-dimensional normed spaces without small-norm hyperplane projections

We prove that there exists an absolute constant $c>1$ such that for every $n\ge 3$ there exists an $n$-dimensional real normed space $X$ for which every linear projection $P$ onto a hyperplane satisfies $\mathrm{\parallel }P\mathrm{\parallel }\ge c$. More generally, for every integer $k\ge 1$ there exists a constant $c_k>1$ such that for every $n\ge k+2$ there exists an $n$-dimensional real normed space $X$ such that every projection $P$ onto a subspace of codimension at most $k$ satisfies $\mathrm{\parallel }P\mathrm{\parallel }\ge c_k$. The proof is based on an idea of Gleit and Lazar, who established an analogous result for the space of real $n\times n$ matrices equipped with either the trace-class norm or the operator norm