  For an interpolation pair $(E_0,E_1)$ of Banach spaces with $E_1
\hookrightarrow E_0$ we use vectors $b_1,b_2,\ldots \in E_1$ that satisfy an
extremal property with respect to the $J$- and $K$-functional to construct
sub-spaces that are isometric to $\ell_q^{(\theta)}$. The construction is based
on a randomisation using independent Rademacher variables. We verify that
systems obtained by re-scaling a function with a certain periodicity property
share this extreme property. This implies the sharpness of natural embeddings
of H\"older spaces obtained by the real interpolation into the corresponding
Gaussian Besov spaces.
