On the best approximation of set-valued functions

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 subset of C(M) be a finite dimensional subspace of C(M). The problem of the Chebyshev approximation of a function f is an element of C(M) by functions from V can be put in the form max(t is an element of M) max{f(t) - g(t), g(t) - f(t)} --> min, g is an element of V. In this paper we solve the two optimization problems max(t is an element of M)max{sigma(+)(t) - g(t), g(t)-sigma(-)(t)} --> min, g is an element of V and max(t is an element of M)max{sigma(-)(t) -g(t), g(t) -sigma(+) (t)} --> min, g is an element of V, where both the functions -sigma(-), sigma(+) : M --> IR are upper and lower semicontinuous, respectively, and satisfy sigma(-)(t) less than or equal to sigma(+)(t) for each t is an element of M. Both the problems can be interpreted as Chebyshev approximation of the set-valued function Sigma : M --> IR with Sigma(t) = [sigma(-)(t), sigma(+)(t)] 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.