Approximation of Planar Convex Sets from Hyperplane Probes

Abstract

This paper studies a geometric probing problem. Suppose that an unknown convex set in R 2 can be probed by an oracle which, when given a unit vector, will return the position of the supporting hyperplane of the convex set that has the given vector as an outward normal. We present an on-line algorithm for choosing probing directions so that, after n probes, an inner and an outer estimate of the convex set are obtained that are within $O(n^{-2})$ of each other in Hausdorff distance. This is optimal since there exist convex sets that, even if visible, cannot be approximated better than $O(n^{-2})$ with n-sided polygons, for example, a circle.