Components of the Kernel in a Staircase Starshaped Polygon

For a non-simply connected orthogonal polygon \( T \), assume that \( T = S \setminus (A_1 \cup \ldots \cup A_n) \), where \( S \) is a simply connected orthogonal polygon and where \( A_1, \ldots, A_n \) are pairwise disjoint sets, each the connected interior of an orthogonal polygon, \( A_i \subset S, 1 \leq i \leq n \). If set \( T \) is staircase starshaped, then \( \text{Ker } T = \bigcap \{\text{Ker } (S \setminus A_i) : 1 \leq i \leq n\} \). Moreover, each component of this kernel will be the intersection of the nonempty staircase convex set \( \text{Ker } S \) with a box, providing an easy proof that each of these components is staircase convex. Finally, there exist at most \( (n + 1)^2 \) such components, and the bound \( (n + 1)^2 \) is best possible.