Generalized Steiner selections applied to standard problems of set-valued numerical analysis. (English) Zbl 1149.28007

Baier, Robert

Staicu, Vasile (ed.), Differential equations, chaos and variational problems. Papers from the conference “Views on ODE’s”, Aveiro, Portugal, June 2006. Basel: Birkhäuser (ISBN 978-3-7643-8481-4/hbk). Progress in Nonlinear Differential Equations and Their Applications 75, 49-60 (2008).

The paper presents a series of results providing a background for developing numerical methods of evaluation of the Aumann integral of a set-valued mapping using generalized Steiner selectors.
A generalized Steiner point of a convex compact set in the Euclidean space is the result of integration of the support set as a function of the unit normal vector over the unit sphere, using an arbitrary Borel measure on the sphere. For a convex-valued mapping and a fixed measure, the generalized Steiner point evaluated pointwisely defines a selector. The evaluation of a generalized Steiner point or a selector is commutable with many set operations, including the evaluation of the Aumann integral of a set-valued mapping. A dense subset of Aumann’s integral can be constructed using a countable set of generalized Steiner selectors generated by continuous measures. For a mapping Lipschitz-continuous with respect to Demyanov’s distance, the author has obtained a linear convergence estimate for the approximation of an Aumann integral by a quadrature formula.
For the entire collection see [Zbl 1131.34002].

Reviewer: Dmitry Silin (Berkeley)

MSC:

28B20	Set-valued set functions and measures; integration of set-valued functions; measurable selections
54C60	Set-valued maps in general topology
65D30	Numerical integration

Keywords:

generalized {Steiner} point; set-valued mapping; quadrature formula; Demyanov distance

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