A monotone path-connected set with outer radially lower continuous metric projection is a strict sun

A monotone path-connected set is known to be a sun in a finite-dimensional Banach space. We show that a $B$-sun (a set whose intersection with each closed ball is a sun or empty) is a sun. We prove that in this event a $B$-sun with ORL-continuous (outer radially lower continuous) metric projection is a strict sun. This partially converses one well-known result of Brosowski and Deutsch. We also show that a $B$-solar LG-set (a global minimizer) is a $B$-connected strict sun.