  Let $({\mathbf X},\|\cdot\|_{\mathbf X}), ({\mathbf Y},\|\cdot\|_{\mathbf
Y})$ be normed spaces with ${\mathrm{dim}}({\mathbf X})=n$. Bourgain's almost
extension theorem asserts that for any ${\varepsilon}>0$, if ${\mathcal{N}}$ is
an ${\varepsilon}$-net of the unit sphere of ${\mathbf X}$ and
$f:{\mathcal{N}}\to {\mathbf Y}$ is $1$-Lipschitz, then there exists an
$O(1)$-Lipschitz $F:{\mathbf X}\to {\mathbf Y}$ such that
$\|F(a)-f(a)\|_{\mathbf Y}\lesssim n{\varepsilon}$ for all $a\in \mathcal{N}$.
We prove that this is optimal up to lower order factors, i.e., sometimes
$\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\gtrsim
n^{1-o(1)}{\varepsilon}$ for every $O(1)$-Lipschitz $F:{\mathbf X}\to {\mathbf
Y}$. This improves Bourgain's lower bound of $\max_{a\in {\mathcal{N}}}
\|F(a)-f(a)\|_{\mathbf Y}\gtrsim n^{c}{\varepsilon}$ for some $0<c<\frac12$. If
${\mathbf X}=\ell_2^n$, then the approximation in the almost extension theorem
can be improved to $\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\lesssim
\sqrt{n}{\varepsilon}$. We prove that this is sharp, i.e., sometimes
$\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\gtrsim
\sqrt{n}{\varepsilon}$ for every $O(1)$-Lipschitz $F:\ell_2^n\to {\mathbf Y}$.
