Sets Expressible as Unions of Staircase n -Convex Polygons

Marilyn Breen

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2011)

  • Volume: 50, Issue: 1, page 23-28
  • ISSN: 0231-9721

Abstract

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Let k and n be fixed, k 1 , n 1 , and let S be a simply connected orthogonal polygon in the plane. For T 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 i k .

How to cite

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Breen, Marilyn. "Sets Expressible as Unions of Staircase $n$-Convex Polygons." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 50.1 (2011): 23-28. <http://eudml.org/doc/197243>.

@article{Breen2011,
abstract = {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 $P_i$, $1 \le i \le k$.},
author = {Breen, Marilyn},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {orthogonal polygons; staircase $n$-convex polygons; orthogonal polygons; staircase -convex polygons},
language = {eng},
number = {1},
pages = {23-28},
publisher = {Palacký University Olomouc},
title = {Sets Expressible as Unions of Staircase $n$-Convex Polygons},
url = {http://eudml.org/doc/197243},
volume = {50},
year = {2011},
}

TY - JOUR
AU - Breen, Marilyn
TI - Sets Expressible as Unions of Staircase $n$-Convex Polygons
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2011
PB - Palacký University Olomouc
VL - 50
IS - 1
SP - 23
EP - 28
AB - 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 $P_i$, $1 \le i \le k$.
LA - eng
KW - orthogonal polygons; staircase $n$-convex polygons; orthogonal polygons; staircase -convex polygons
UR - http://eudml.org/doc/197243
ER -

References

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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. (2006) Zbl1157.52303MR2194749
  2. Breen, M., 10.1007/s00605-005-0345-9, Monatsh. Math. 148 (2006), 91–100. (2006) Zbl1134.52007MR2235357DOI10.1007/s00605-005-0345-9
  3. Breen, M., 10.1007/BF01189893, Arch. Math. 63 (1994), 182–190. (1994) Zbl0742.52006MR1289301DOI10.1007/BF01189893
  4. Breen, M., 10.1007/BF01264043, . Geometriae Dedicata 53 (1994), 49–56. (1994) Zbl0814.52002MR1299884DOI10.1007/BF01264043
  5. Danzer, L., Grünbaum, B., Klee, V., 10.1090/pspum/007/0157289, In: Convexity, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI 7 (1962), 101–180. (1962) MR0157289DOI10.1090/pspum/007/0157289
  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. (1993) Zbl0791.52009MR1242986
  7. Hare, W. R., Jr., Kenelly, J. W., 10.1090/S0002-9939-1970-0257879-7, Proc. Amer. Math. Soc. 25 (1970), 379–380. (1970) Zbl0195.51603MR0257879DOI10.1090/S0002-9939-1970-0257879-7
  8. Lawrence, J. F., Hare, W. R., Jr., Kenelly, J. W., 10.1090/S0002-9939-1972-0291952-4, Proc. Amer. Math. Soc. 34 (1972), 225–228. (1972) Zbl0237.52001MR0291952DOI10.1090/S0002-9939-1972-0291952-4
  9. Lay, S. R., Convex Sets and Their Applications, John Wiley, New York, 1982. (1982) Zbl0492.52001MR0655598
  10. McKinney, R. L., 10.4153/CJM-1966-088-7, Canad. J. Math 18 (1966), 883–886. (1966) Zbl0173.15305MR0202049DOI10.4153/CJM-1966-088-7
  11. Motwani, R., Raghunathan, A., Saran, H., 10.1016/0022-0000(90)90017-F, J. Comput. Syst. Sci. 40 (1990), 19–48. (1990) Zbl0705.68082MR1047288DOI10.1016/0022-0000(90)90017-F
  12. Valentine, F. A., Convex Sets, McGraw-Hill, New York, 1964. (1964) Zbl0129.37203MR0170264

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