Set-valued approximations with Minkowski averages convergence and convexification rates

Nira Dyn, Elza Farkhi*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Three approximation processes for set-valued functions (multifunctions) with compact images in ℝn are investigated. Each process generates a sequence of approximants, obtained as finite Minkowski averages (convex combinations) of given data of compact sets in ℝn. 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.

Original languageEnglish
Pages (from-to)363-377
Number of pages15
JournalNumerical Functional Analysis and Optimization
Issue number3-4
StatePublished - May 2004


  • Bernstein type approximation
  • Convexity
  • Measures of nonconvexity
  • Minkowski addition
  • Riemann sums
  • Set-valued functions
  • Set-valued integral
  • Spline subdivision schemes


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