  Suppose that one observes pairs $(x_1,Y_1)$, $(x_2,Y_2)$, ..., $(x_n,Y_n)$,
where $x_1\le x_2\le ... \le x_n$ are fixed numbers, and $Y_1,Y_2,...,Y_n$ are
independent random variables with unknown distributions. The only assumption is
that ${\rm Median}(Y_i)=f(x_i)$ for some unknown convex function $f$. We
present a confidence band for this regression function $f$ using suitable
multiscale sign-tests. While the exact computation of this band requires
$O(n^4)$ steps, good approximations can be obtained in $O(n^2)$ steps. In
addition the confidence band is shown to have desirable asymptotic properties
as the sample size $n$ tends to infinity.
