Abstract
LetT be a simply connected orthogonal polygon having the property that for every three points ofT, at least two of these points see each other via staircases inT. ThenT is a union of three orthogonally convex polygons. The number three is best possible.
ForT a simply connected orthogonal polygon,T is a union of two orthogonally convex polygons if and only if for every sequencev 1,...,v n,v n+1 =v 1 inT, n odd, at least one consecutive pairv i ,v i+1 sees each other via staircase paths inT, 1 ≤i ≤n. An analogous result says thatT is a union of two orthogonal polygons which are starshaped via staircase paths if and only if for every odd sequence inT, at least one consecutive pair sees a common point via staircases inT.
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Supported in part by NSF grants DMS-8908717 and DMS-9207019.
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Breen, M. Unions of orthogonally convex or orthogonally starshaped polygons. Geom Dedicata 53, 49–56 (1994). https://doi.org/10.1007/BF01264043
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DOI: https://doi.org/10.1007/BF01264043