Covering a Set by a Convex Compactum: Error Estimates and Computation

Abstract

Abstract: A problem related to that of finding the Chebyshev center of a compact convex set in $$\mathbb R^n$$ is considered, namely, the problem of calculating the center and the least positive ratio of a homothety under which the image of a given compact convex set in $$\mathbb R^n$$ covers another given compact convex set. Both sets are defined by their support functions. A solution algorithm is proposed which consists in discretizing the support functions on a grid of unit vectors and reducing the problem to a linear programming problem. The error of the solution is estimated in terms of the distance between the given set and its approximation in the Hausdorff metric. For the stability of the approximate solution, it is essential that the sets be uniformly convex and a certain set in the dual space has a nonempty interior.