  In 1931 Bohnenblust and Hille proved that for each m-homogeneous polynomial
$\sum_{|\alpha| = m} a_\alpha z^\alpha$ on $\C^n$ the
$\ell^{\frac{2m}{m+1}}$-norm of its coefficients is bounded from above by a
constant $C_m$ (depending only on the degree $m$) times the sup norm of the
polynomial on the polydisc $\mathbb{D}^n$. We prove that this inequality is
hypercontractive in the sense that the optimal constant $C_m$ is $\leq C^m$
where $C \geq 1$ is an absolute constant. From this we derive that the Bohr
radius $K_n$ of the $n$-dimensional polydisc in $\mathbb{C}^n$ is up to an
absolute constant $\geq \sqrt{\log n/n}$; this result was independently and
with a differnt proof discovered by Ortega-Cerd{\`a}, Ouna\"ies and Seip. An
alternative approach even allows to prove that the Bohr radius $K_n^p$, $1 \leq
p \leq \infty $ of the unit ball of $\ell_n^p ,$ is asymptotically $ \geq (\log
n/n) ^{1-1/ \min (p,2)}$. This shows that the upper bounds for $K_n^p$ given by
Boas and Khavinson are optimal.
