Nonexistence of continuous selections of the metric projection for a class of weak Chebyshev spaces

Abstract:

The class $\mathfrak {B}$ of all those n-dimensional weak Chebyshev subspaces of $C [a, b]$ whose elements have no zero intervals is considered. It is shown that there does not exist any continuous selection of the metric projection for G if there is a nonzero g in G having at least $n + 1$ distinct zeros. Together with a recent result of Nürnberger-Sommer, the following characterization of continuous selections for $\mathfrak {B}$ is valid: There exists a continuous selection of the metric projection for G in $\mathfrak {B}$ if and only if each nonzero g in G has at most n distinct zeros.