Suitable families of boxes and kernels of staircase starshaped sets in \(\mathbb{R}^d\) (Q2248825)

Let \(S\subset \mathbb{R}^d\) be an orthogonal polytope and let \(C\) be a suitable family of boxes for \(S\). The author proves two theorems. The first one allows to recognize points in the kernel of an orthogonal polytope. Namely, if \(\text{Ker } S\neq \emptyset\) then it is a union of boxes in \(C\). Moreover, for a box \(B\) in \(C\) the necessary and sufficient conditions are provided in order to have \(B\subset \text{Ker } S\). Now let \(p\in S\). The second theorem provides the necessary and sufficient conditions in terms of paths \(\lambda(p, x)\), \(x\in S\), in order to have \(p\in \text{Ker } S\). This strengthens Theorem 1 from [the author, Beitr. Algebra Geom. 51, No. 1, 251--261 (2010; Zbl 1204.52008)].