On staircase starshapedness in rectilinear spaces (Q1126447)

Let \(P\) be a simple rectilinear polyhedron in \(n\)-dimensional space. A staircase path inside \(P\) is a monotone rectilinear path inside \(P\). If every two boundary points of \(P\) can be connected via staircase paths from a common point of \(P\), then \(P\) is starshaped via staircase paths. This extends the same result in the plane, which was known before. The same result holds true when \(P\) is a cubical polyhedron, which is the geometric realization of some median graph. For nonsimple rectilinear polyhedra the situation is probably much more difficult.