Points of Monotone Connectedness and Solar Properties of Sets

Abstract

Luminosity points of sets are studied; these are the points for which the Kolmogorov (solar) condition for a best approximant holds. We obtain sufficient conditions for a given point of a set to be a luminosity point of this set. The results are formulated in terms of points of monotone connectedness of the set. We consider the characteristics of luminosity points which involve individual characteristics of both the space and the set (points of monotone connectedness of a set and rotund points of the unit sphere of the space). Prox-monotone path-connected sets are introduced; for such a set \(M\), if a point \(y\) is a nearest point from this set for some point \(x\notin M\), then \(y\) is a point of monotone connectedness of \(M\). We also show that each boundedly compact prox-monotone path-connected set (prox-Menger connected set) is a sun. Applications to the space \(c_0\) are also given.