Approximation of Set-Valued Functions: Adaptation of Classical Approximation Operators.pdf
This book is aimed at the approximation of set-valued functions with compact sets in an Euclidean space as values. The interest in set-valued functions is rather new. Such functions arise in various modern areas such as control theory, dynamical systems and optimization. The authors' motivation also comes from the newer field of geometric modeling, in particular from the problem of reconstruction of 3D objects from 2D cross-sections. This is reflected in the focus of this book, which is the approximation of set-valued functions with general (not necessarily convex) sets as values, while previous results on this topic are mainly confined to the convex case. The approach taken in this book is to adapt classical approximation operators and to provide error estimates in terms of the regularity properties of the approximated set-valued functions. Specialized results are given for functions with 1D sets as values.
Scientific Background: On Functions with Values in Metric Spaces: Basic Notions, Approximation Operators; On Sets: Sets, Set Operations and Parametrizations; On Set-Valued Functions (SVFs): Representations and Regularity; Approximation of SVFs: Methods Based on Canonical Representations; Methods Based on Minkowski Convex Combinations; Methods Based on the Metric Average; Methods Based on Metric Linear Combinations; Methods Based on Metric Selections; Approximation of SVFs with Images in R: Regularity of the Boundaries of the Graph; Multisegmental and Topological Representations; Methods Based on Topological Representation.