Set-Valued Approximations with Minkowski Averages – Convergence and Convexification Rates

Three approximation processes for set-valued functions (multifunctions) with compact images in ℝ n are investigated. Each process generates a sequence of approximants, obtained as finite Minkowski averages (convex combinations) of given data of compact sets in ℝ n . The limit of the sequence exists and and is equal to the limit of the same process, starting from the convex hulls of the given data. The common phenomenon of convexification of the approximating sequence is investigated and rates of convergence are obtained. The main quantitative tool in our analysis is the Pythagorean type estimate of Cassels for the “inner radius” measure of nonconvexity of a compact set. In particular we prove the convexity of the images of the limit multifunction of set-valued spline subdivision schemes and provide error estimates for the approximation of set-valued integrals by Riemann sums of sets and for Bernstein type approximation to a set-valued function.