  We consider the problem of isotonic regression, where the underlying signal
$x$ is assumed to satisfy a monotonicity constraint, that is, $x$ lies in the
cone $\{ x\in\mathbb{R}^n : x_1 \leq \dots \leq x_n\}$. We study the isotonic
projection operator (projection to this cone), and find a necessary and
sufficient condition characterizing all norms with respect to which this
projection is contractive. This enables a simple and non-asymptotic analysis of
the convergence properties of isotonic regression, yielding uniform confidence
bands that adapt to the local Lipschitz properties of the signal.
