Rates of convergence for the sup-norm risk in image models under sequential designs

Let G be that portion of the unit square which lies below the graph of a smooth function. Assume that observations of the indicator function of G are available at any points X1,...,Xn in the plane. If each consecutive point Xi can be chosen sequentially, on the basis of all the preceding data, then how accurately can the smooth function be estimated in sup-norm? Using the boundary fragment model, this question translates into a question about the large sample performance of the minimax risks under sup-norm loss. The asymptotic rates of these risks are found. The results are extended to additive noise models and multidimensional images.