A ball of radius \(\varepsilon \) has diameter at most \(2\varepsilon \), so if \(2\varepsilon {\lt} \delta \) then each ball of an admissible cover contains at most one of the points. Assigning to every point a covering centre is then injective, so there are at least as many centres as points — more than \(2^n\), contradicting admissibility.

Both are built one vector at a time. For \(d_n\), take \(V_j\) to be the span of the images already chosen — of dimension at most \(n\) — and pick \(x_j \in B_X\) with \(\| S x_j + w\| {\gt} \gamma \) for all \(w \in V_j\), which is possible because \(\gamma {\lt} d_n(S)\) and \(\dim V_j \le n\) (Definition 3.7). For \(c_n\), make the same selection below \(c_n(S)\) — using \(\gamma {\lt} c_n(S)\) and Definition 3.5 — on the common kernel of the functionals already chosen, a closed subspace of codimension at most \(n\), and rescale the resulting unit-image vector by \(\gamma \) to land in \(B_X\).

Let \(\gamma {\lt} d_n(S)\) and take a flag \(x_0, \dots , x_n\) as in Theorem 11.6. For \(\sigma \in \{ \pm 1\} ^{n+1}\) set

For \(\sigma \neq \tau \) let \(j\) be the largest index with \(\sigma _j \neq \tau _j\). Then \(z_\sigma - z_\tau = \tfrac {1}{n+1}\bigl( (\sigma _j - \tau _j) S x_j + w \bigr)\) with \(w \in V_j\), because the indices \(k {\gt} j\) contribute nothing and those \(k {\lt} j\) contribute elements of \(V_j\). Since \(|\sigma _j - \tau _j| = 2\), the flag property gives \(\| z_\sigma - z_\tau \| {\gt} 2\gamma /(n+1)\). So \(2^{n+1}\) points of \(S(B_X)\) are pairwise more than \(2\gamma /(n+1)\) apart, and Proposition 11.5 yields \(e_n(S) \ge \gamma /(n+1)\). Letting \(\gamma \nearrow d_n(S)\) gives the claim.

For \(c_n\) the same computation is read off by the functional \(\rho _j\) at the smallest disagreeing index \(j\): it annihilates \(S x_k\) for \(k {\gt} j\), while \(\sigma _k = \tau _k\) kills the terms with \(k {\lt} j\).