  In this paper, we consider adaptive estimation of an unknown planar compact,
convex set from noisy measurements of its support function on a uniform grid.
Both the problem of estimating the support function at a point and that of
estimating the convex set are studied. Data-driven adaptive estimators are
proposed and their optimality properties are established. For pointwise
estimation, it is shown that the estimator optimally adapts to every compact,
convex set instead of a collection of large parameter spaces as in the
conventional minimax theory of nonparametric estimation. For set estimation,
the estimators adaptively achieve the optimal rate of convergence. In both
these problems, our analysis makes no smoothness assumptions on the unknown
sets.
