Аннотация:Generalized n-piecewise functions constructed from givenmonotone path-connected boundedly compact subsets of the space C[a, b]are studied. They are shown to be monotone path-connected suns. Infinite-dimensional polyhedral spaces, luminosity points of sets admittinga lower semicontinuous selection of the metric projection operator are investigated.An example of a non-B-connected sun in a four-dimensional polyhedralnormed space is constructed.