A Krasnosel’skiltype Theorem for an Orthogonal Polytope Whose Complement Contains Open Boxes

Let \( T = S \setminus \left( \cup \left\{ A : A \, \text{ in } \, \mathcal{A} \right\} \right) \), where \( S \) is an orthogonal polytope in \( \mathbb{R}^d \) for \( d \geq 2 \) and where \( \mathcal{A} \) is a collection of \( n \) pairwise disjoint open boxes contained in \( S \). Point \( x \) belongs to \(\text{Ker } T \) if and only if \( x \) belongs to \(\text{Ker } S \) and no coordinate line at \( x \) meets any \( A \) in \( \mathcal{A} \). In turn, this relationship between the staircase kernels of \( S \) and \( T \) produces a Krasnosel’skii-type result for \( T \) in terms of \( n \), extending the class of orthogonal polytopes for which such a theorem exists.