Solarity and connectedness of sets in the space $C[a,b]$ and in finite-dimensional polyhedral spaces

Generalized $n$-piecewise functions constructed from given monotone path-connected boundedly compact subsets of the space $C[a,b]$ are studied. They are shown to be monotone path-connected suns. In finite-dimensional polyhedral spaces, luminosity points of sets admitting a lower semicontinuous selection of the metric projection operator are investigated. An example of a non-$B$-connected sun in a four-dimensional polyhedral normed space is constructed.
Bibliography: 14 titles.