There is an error in the proof of Corollary 7.8; there is no reason why the functionals $\psi_j$ should exist as claimed.

However, the results $\tau^*(X)\le 2d_{\rm B}(X)$ does hold.

To see this, if $d>d_{\rm B}(X)$ then we can cover $Z:=X-X$ with $\epsilon$-balls centred at $z_j$; there are $\le C\epsilon^{-2d}$ of these.

Now use the Hahn-Banach Theorem to construct linear functions $\psi_j$ with $\|\psi_j\|=1$ and $\psi_j(z_j)=\|z-j\|$; let $V$ be the subsapce of $B^*$ spanned by the $\{\psi_j\}$.

Take $z\in Z$ with $\|z\|\ge 3\epsilon$. Then there exists a $z_k$ such that $\|z-z_k\|<\epsilon$ and so, since $\|z_k\|>2\epsilon$, we have

$|\psi_k(z)|=|\psi_k(z-z_k)+\psi_k(z_k)|\ge\|z_k\|-\|z-z_k\|>\epsilon$.

This shows that $\tau^*(X)\le\sigma_{1/3}(X)\le 2d_{\rm B}(X)$.

Lemma 9.2, it should be $(2^NN^{N/2},N)$ homogeneous. In the second line the cubes should have side $\rho$ and the balls radius $\sqrt{N}\,\rho/2$; the bound on the number of balls is for a cover by balls of radius $\sqrt{N}\,\rho/2$, which leads to the correct result.