Local solarity of suns in normed linear spaces

The paper is concerned with solarity of intersections of suns with bars (in particular, with closed balls and extreme hyperstrips) in normed linear spaces. A sun in a finite-dimensional $(BM)$-space (in particular, in $\ell^1(n)$) is shown to be monotone path connected. A nonempty intersection of an $\mathrm m$-connected set (in particular, a sun in a two-dimensional space or in a finite-dimensional $(BM)$-space) with a bar is shown to be a monotone path-connected sun. Similar results are obtained for boundedly compact subsets of infinite-dimensional spaces. A nonempty intersection of a monotone path-connected subset of a normed space with a bar is shown to be a monotone path-connected $\alpha$-sun.