  The norm of the Riesz projection from $L^\infty(\T^n)$ to $L^p(\T^n)$ is
considered. It is shown that for $n=1$, the norm equals $1$ if and only if
$p\le 4$ and that the norm behaves asymptotically as $p/(\pi e)$ when $p\to
\infty$. The critical exponent $p_n$ is the supremum of those $p$ for which the
norm equals $1$. It is proved that $2+2/(2^n-1)\le p_n <4$ for $n>1$; it is
unknown whether the critical exponent for $n=\infty$ exceeds $2$.
