$L_1$-optimal linear programming estimator for periodic frontier functions with Hölder continuous derivative

We propose a new estimator based on a linear programming method for smooth frontiers of sample points on a plane. The derivative of the frontier function is supposed to be Hölder continuous. The estimator is defined as a linear combination of kernel functions being sufficiently regular, covering all the points and whose associated support is of smallest surface. The coefficients of the linear combination are computed by solving a linear programming problem. The $L_1$ error between the estimated and the true frontier function is shown to be almost surely converging to zero, and the rate of convergence is proved to be optimal.