Components of the Kernel in a Staircase Starshaped Polygon

Abstract

For a non-simply connected orthogonal polygon $T$, assume that $T=S\setminus (A_1\cup \dots \cup A_n)$, where $S$ is a simply connected orthogonal polygon and where $A_1,\dots ,A_n$ are pairwise disjoint sets, each the connected interior of an orthogonal polygon, $A_i\subset S,1\le i\le n$. If set $T$ is staircase starshaped, then $\text{Ker }T=\bigcap \{\text{Ker }(S\setminus A_i):1\le i\le n\}$. 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.