Adaptive Estimation of Planar Convex Sets

In this paper, we consider adaptive estimation of an unknown planar compact, convex set from noisy measurements of its support function. Both the problem of estimating the support function at a point and that of estimating the whole convex set are studied. For pointwise estimation, we consider the problem in a general non-asymptotic framework, which evaluates the performance of a procedure at each individual set, instead of the worst case performance over a large parameter space as in conventional minimax theory. A data-driven adaptive estimator is proposed and is shown to be optimally adaptive to every compact, convex set. For estimating the whole convex set, we propose estimators that are shown to adaptively achieve the optimal rate of convergence. In both these problems, our analysis makes no smoothness assumptions on the boundary of the unknown convex set.