On the Best Approximation of Set-Valued Functions

Abstract

Let M be a Hausdorff compact topological space, let C(M) be the Banach space of the continuous on M functions supplied with the supremum norm and let V C C(M) be a finite dimensional subspace of C(M). The problem of the Chebyshev approximation of a function f ∈ C(M) by functions from V can be put in the form \({{\max }_{{t \in M}}}\max \left\{ {f\left( t \right) - g\left( t \right),g\left( t \right) - f\left( t \right)} \right\} \to \min , \) g ∈ V. In this paper we solve the two optimization prob-lems \({{\max }_{{t \in M}}}\max \left\{ {{{\sigma }^{ + }}\left( t \right) - g\left( t \right),g\left( t \right) - {{\sigma }^{ - }}\left( t \right)} \right\} \to \min , \) g∈ V and \({{\max }_{{t \in M}}}\max \left\{ {{{\sigma }^{ + }}\left( t \right) - g\left( t \right),g\left( t \right) - {{\sigma }^{ - }}\left( t \right)} \right\} \to \min ,\) g∈ V, where both the functions \(- {{\sigma }^{ - }},{{\sigma }^{ + }}:M \to \mathbb{R}\) are upper and lower semicontinuous, respectively, and satisfy \({{\sigma }^{ - }}\left( t \right) \leqslant {{\sigma }^{ + }}\left( t \right) \) for each t ∈ M. Both the problems can be interpreted as Chebyshev approximation of the set-valued function \( \sum : M \to \mathbb{R}with\sum {\left( t \right)} = \left[ {{{\sigma }^{ - }}\left( t \right),{{\sigma }^{ + }}\left( t \right)} \right]\) using suitable distances between a point and a set. The first problem occur e.g. in curve fitting with noisy data or in approximating spatial bodies by circular cylinders with respect to a proper distance. The second problem is useful for calculating continuous selections with special uniform distance properties.