Maximal visibility and unions of orthogonally starshaped sets

Abstract

Let S be an orthogonal polygon in the plane. For each point x in S,let $V_x$ denote the set of points which x sees via staircase paths and let $M_x = {y : V_y = V_x}$. For S simply connected, S is starshaped via staircase paths (i.e., orthogonally starshaped) if and only if S contains exactly one such closed set $M_x$, and when this occurs $M_x$ is the staircase kernel of S. In general, if S contains exactly k such distinct closed set $M_{x₁},...M_{x_k}$, then S is a union of k (or possibly fewer) orthogonally starshaped sets chosen from $V_{x₁},...,V_{x_k}$.