A Krasnosel'skii-type theorem for certain orthogonal polytopes starshaped via \(k\)-staircase paths (Q2174104)

The author considers a set \(S\) that is a finite union of boxes in \(\mathbb{R}^{d}\) whose intersection graph is a tree. Few fundamental results are extended taking into account visibility by means of \(k\)-staircase paths, paths that are unions of \(k\) (or fewer) staircase paths. The new theorems are: a \(k\) path analogue of the main result of \textit{V. Chepoi} [Geom. Dedicata 63, No. 3, 321--329 (1996; Zbl 0866.52006)], a generalization of a previous result of result of the author from 2010 [ibid. 17, No. 4, 525--532 (2017; Zbl 1410.52005)] and a generalization of the classic \textit{M. A. Krasnosel'skij} characterization theorem [Mat. Sb., Nov. Ser. 19(61), 309--310 (1946; Zbl 0061.37705)] of starshaped sets by a visibility property. It is proved that, if every two points of \(S\) see a common point via \(k\)-staircase paths, then \(S\) will be starshaped via \(k\)-staircase paths. More, it is proved that the \(k\)-staircase kernel of \(S\) is convex via \(k\)-staircase paths.