Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Illumination for unions of boxes in $ {\bf R}\sp d$


Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 116 (1992), 197-202
MSC: Primary 52A30
DOI: https://doi.org/10.1090/S0002-9939-1992-1089402-6
MathSciNet review: 1089402
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ S$ be a finite union of boxes (polytopes whose edges are parallel to the coordinate axes) in $ {R^d}$. If every two vertices of $ S$ are clearly illumined by some common translate of the box $ T$, then there is a translate of $ T$ that clearly illumines every point of $ S$ . A similar result holds when appropriate boundary points of $ S$ are illumined (rather than clearly illumined) by translates of box $ T$.


References [Enhancements On Off] (What's this?)

  • [1] A. Bezdek, K. Bezdek, and T. Bistriczky, On illumination in the plane by line segments, Geom. Dedicata (to appear).
  • [2] Marilyn Breen, Illumination by translates of convex sets, Geom. Dedicata (submitted). MR 1163714 (93h:52006)
  • [3] Ludwig Danzer and Branko Grünbaum, Intersection properties of boxes in $ {R^d}$, Combinatorica 2 (1982), 237-246. MR 698651 (84g:52014)
  • [4] Ludwig Danzer, Branko Grünbaum, and Victor Klee, Helly's theorem and its relatives, Convexity, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc, Providence, RI, 1962, pp. 101-180.
  • [5] E. Helly, Über mengen konvexer Körper mit gemeinschaftlichen Punkten, Jahresber. Deutsch. Math. Verein. 32 (1923), 175-176.
  • [6] V. L. Klee, The critical set of a convex body, Amer. J. Math. 75 (1953), 178-188. MR 0052803 (14:678f)
  • [7] M. A. Krasnosel'skii, Sur un critère pour qu'un domaine soit étoilé, Mat. Sb. (61) 19 (1946), 309-310. MR 0020248 (8:525a)
  • [8] Steven R. Lay, Convex sets and their applications, Wiley, New York, 1982. MR 655598 (83e:52001)
  • [9] W. Lenhart, R. Pollack, J. Sack, R. Seidel, M. Sharir, S. Suri, G. Toussaint, S. Whitesides, and C. Yap, Computing the link center of a simple polygon, Discrete Comput. Geom. 3 (1988), 281-293. MR 937288 (89k:68145)
  • [10] Joseph O'Rourke, Art gallery theorems and algorithms, Oxford Univ. Press, 1987. MR 921437 (89f:68067)
  • [11] Godfried Toussaint and Hossam El-Gindy, Traditional galleries are star-shaped if every two paintings are visible from some common point, Technical Report SOCS-81.10, McGill Univ., March 1981.
  • [12] F. A. Valentine, Convex sets, McGraw-Hill, New York, 1964. MR 0170264 (30:503)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52A30

Retrieve articles in all journals with MSC: 52A30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1089402-6
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society