  We consider the problem of nonparametric estimation of a convex regression
function $\phi_0$. We study the risk of the least squares estimator (LSE) under
the natural squared error loss. We show that the risk is always bounded from
above by $n^{-4/5}$ modulo logarithmic factors while being much smaller when
$\phi_0$ is well-approximable by a piecewise affine convex function with not
too many affine pieces (in which case, the risk is at most $1/n$ up to
logarithmic factors). On the other hand, when $\phi_0$ has curvature, we show
that no estimator can have risk smaller than a constant multiple of $n^{-4/5}$
in a very strong sense by proving a "local" minimax lower bound. We also study
the case of model misspecification where we show that the LSE exhibits the same
global behavior provided the loss is measured from the closest convex
projection of the true regression function. In the process of deriving our risk
bounds, we prove new results for the metric entropy of local neighborhoods of
the space of univariate convex functions. These results, which may be of
independent interest, demonstrate the non-uniform nature of the space of
univariate convex functions in sharp contrast to classical function spaces
based on smoothness constraints.
