A decomposition into boxes that determine the Kernel of a staircase starshaped set in \(\mathbb{R}^d\)

Let \( S \) be a connected union of finitely many \( d \)-dimensional boxes in \( \mathbb{R}^d \) and let \( \mathcal{B} \) represent the family of boxes determined by facet hyperplanes for \( S \), with \( \mathcal{F} \) the associated family of faces (including members of \( \mathcal{B} \)). For set \( F \) in \( \mathcal{F} \), point \( x \) relatively interior to \( F \), and point \( y \) in \( S \), \( x \) sees \( y \) via staircase paths in \( S \) if and only if every point of \( F \) sees \( y \) via such paths. Thus the visibility set of \( x \) is a union of members of \( \mathcal{F} \), as is the staircase kernel of \( S \). A similar result holds for \( k \)-staircase paths in \( S \) and the \( k \)-staircase kernel of \( S \).