  Estimation of convex functions finds broad applications in engineering and
science, while convex shape constraint gives rise to numerous challenges in
asymptotic performance analysis. This paper is devoted to minimax optimal
estimation of univariate convex functions from the H\"older class in the
framework of shape constrained nonparametric estimation. Particularly, the
paper establishes the optimal rate of convergence in two steps for the minimax
sup-norm risk of convex functions with the H\"older order between one and two.
In the first step, by applying information theoretical results on probability
measure distance, we establish the minimax lower bound under the supreme norm
by constructing a novel family of piecewise quadratic convex functions in the
H\"older class. In the second step, we develop a penalized convex spline
estimator and establish the minimax upper bound under the supreme norm. Due to
the convex shape constraint, the optimality conditions of penalized convex
splines are characterized by nonsmooth complementarity conditions. By
exploiting complementarity methods, a critical uniform Lipschitz property of
optimal spline coefficients in the infinity norm is established. This property,
along with asymptotic estimation techniques, leads to uniform bounds for bias
and stochastic errors on the entire interval of interest. This further yields
the optimal rate of convergence by choosing the suitable number of knots and
penalty value. The present paper provides the first rigorous justification of
the optimal minimax risk for convex estimation under the supreme norm.
