Geodesic
Approximation of Univariate Set-Valued Functions - an Overview
Serdica Mathematical Journal, Tome 33 (2007) no. 4, pp. 495-514.

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The paper is an updated survey of our work on the approximation of univariate set-valued functions by samples-based linear approximation operators, beyond the results reported in our previous overview. Our approach is to adapt operators for real-valued functions to set-valued functions, by replacing operations between numbers by operations between sets. For set-valued functions with compact convex images we use Minkowski convex combinations of sets, while for those with general compact images metric averages and metric linear combinations of sets are used. We obtain general approximation results and apply them to Bernstein polynomial operators, Schoenberg spline operators and polynomial interpolation operators.
Keywords: Compact Sets, Set-Valued Functions, Linear Approximation Operators, Minkowski Sum of Sets, Metric Average, Metric Linear Combinations 
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     title = {Approximation of {Univariate} {Set-Valued} {Functions} - an {Overview}},
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Dyn, Nira; Farkhi, Elza; Mokhov, Alona. Approximation of Univariate Set-Valued Functions - an Overview. Serdica Mathematical Journal, Tome 33 (2007) no. 4, pp. 495-514. http://geodesic.mathdoc.fr/item/SMJ2_2007_33_4_a6/