Sorry, I can't find a good tag for this.
So the "scaled" $\mathrm{L}^p$ norm is:
$$ \| \mathbf{x} \| = \left( \frac{1}{n} \sum\limits_{j=1}^{n} |x_j|^p \right)^{1/p} $$
and the geometric mean is:
$$ \overline{\mathbf{x}} = \exp \left( \frac{1}{n} \sum\limits_{j=1}^{n} \log(x_j) \right) $$
Is there an elegant way to express the latter in terms of the former?