Sets expressible as unions as staircase \(n\)-convex polygons. (English) Zbl 1244.52009

Summary: Let \(k\) and \(n\) be fixed, \(k\geq 1\), \(n \geq 1\), 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 \(P_i\), \(1 \leq i \leq k\).


52A35 Helly-type theorems and geometric transversal theory
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[1] Breen, M.: A Helly theorem for intersections of sets starshaped via staircase \(n\)-paths. Ars Combinatoria 78 (2006), 47-63. · Zbl 1157.52303
[2] Breen, M.: Intersections and unions of orthogonal polygons starshaped via staircase \(n\)-paths. Monatsh. Math. 148 (2006), 91-100. · Zbl 1134.52007
[3] Breen, M.: Staircase \(k\)-kernels for orthogonal polygons. Arch. Math. 63 (1994), 182-190. · Zbl 0742.52006
[4] Breen, M.: Unions of orthogonally convex or orthogonally starshaped polygons. Geometriae Dedicata 53 (1994), 49-56. · Zbl 0814.52002
[5] Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. Convexity, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI 7 (1962), 101-180. · Zbl 0132.17401
[6] Eckhoff, J.: Helly, Radon, and Carathéodory type theorems. Gruber, P. M., Wills, J. M., (eds.) Handbook of Convex Geometry, vol. A, North Holland, New York (1993), 389-448. · 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. · 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. · Zbl 0237.52001
[9] Lay, S. R.: Convex Sets and Their Applications. John Wiley, New York, 1982. · Zbl 0492.52001
[10] McKinney, R. L.: On unions of two convex sets. Canad. J. Math 18 (1966), 883-886. · 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. · Zbl 0705.68082
[12] Valentine, F. A.: Convex Sets. McGraw-Hill, New York, 1964. · Zbl 0129.37203
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