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Keywords:
orthogonal polygons; staircase $n$-convex polygons
orthogonal polygons; staircase $n$-convex polygons
Summary:
Let $k$ and $n$ be fixed, $k\ge 1$, $n \ge 1$, and let $S$ be a simply connected orthogonal polygon in the plane. For $T \subseteq S, T$ lies in a staircase $n$-convex orthogonal polygon $P$ in $S$ if and only if every two points of $T$ see each other via staircase $n$-paths in $S$. This leads to a characterization for those sets $S$ expressible as a union of $k$ staircase $n$-convex polygons $P_i$, $1 \le i \le k$.
References:
[1] Breen, M.: A Helly theorem for intersections of sets starshaped via staircase $n$-paths. Ars Combinatoria 78 (2006), 47–63. MR 2194749 | Zbl 1157.52303
[2] Breen, M.: Intersections and unions of orthogonal polygons starshaped via staircase $n$-paths. Monatsh. Math. 148 (2006), 91–100. DOI 10.1007/s00605-005-0345-9 | MR 2235357 | Zbl 1134.52007
[3] Breen, M.: Staircase $k$-kernels for orthogonal polygons. Arch. Math. 63 (1994), 182–190. DOI 10.1007/BF01189893 | MR 1289301 | Zbl 0742.52006
[4] Breen, M.: Unions of orthogonally convex or orthogonally starshaped polygons. Geometriae Dedicata 53 (1994), 49–56. DOI 10.1007/BF01264043 | MR 1299884 | Zbl 0814.52002
[5] Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Convexity, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI 7 (1962), 101–180. DOI 10.1090/pspum/007/0157289 | MR 0157289
[6] Eckhoff, J.: Helly, Radon, and Carathéodory type theorems. In: Gruber, P. M., Wills, J. M., (eds.) Handbook of Convex Geometry, vol. A, North Holland, New York (1993), 389–448. MR 1242986 | Zbl 0791.52009
[7] Hare, W. R., Jr., Kenelly, J. W.: Sets expressible as unions of two convex sets. Proc. Amer. Math. Soc. 25 (1970), 379–380. DOI 10.1090/S0002-9939-1970-0257879-7 | MR 0257879 | Zbl 0195.51603
[8] Lawrence, J. F., Hare, W. R., Jr., Kenelly, J. W.: Finite unions of convex sets. Proc. Amer. Math. Soc. 34 (1972), 225–228. DOI 10.1090/S0002-9939-1972-0291952-4 | MR 0291952 | Zbl 0237.52001
[9] Lay, S. R.: Convex Sets and Their Applications. John Wiley, New York, 1982. MR 0655598 | Zbl 0492.52001
[10] McKinney, R. L.: On unions of two convex sets. Canad. J. Math 18 (1966), 883–886. DOI 10.4153/CJM-1966-088-7 | MR 0202049 | Zbl 0173.15305
[11] Motwani, R., Raghunathan, A., Saran, H.: Covering orthogonal polygons with star polygons: the perfect graph approach. J. Comput. Syst. Sci. 40 (1990), 19–48. DOI 10.1016/0022-0000(90)90017-F | MR 1047288 | Zbl 0705.68082
[12] Valentine, F. A.: Convex Sets. McGraw-Hill, New York, 1964. MR 0170264 | Zbl 0129.37203
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