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 $\textbf {R}^ d$

Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 116 (1992), 197-202
MSC: Primary 52A30
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?)

    A. Bezdek, K. Bezdek, and T. Bistriczky, On illumination in the plane by line segments, Geom. Dedicata (to appear).
  • Marilyn Breen, Illumination by translates of convex sets, Geom. Dedicata 42 (1992), no. 2, 215–222. MR 1163714, DOI
  • Ludwig Danzer and Branko GrĂĽnbaum, Intersection properties of boxes in ${\bf R}^{d}$, Combinatorica 2 (1982), no. 3, 237–246. MR 698651, DOI
  • Ludwig Danzer, Branko GrĂĽnbaum, 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. E. Helly, Ăśber mengen konvexer Körper mit gemeinschaftlichen Punkten, Jahresber. Deutsch. Math. Verein. 32 (1923), 175-176.
  • V. L. Klee Jr., The critical set of a convex body, Amer. J. Math. 75 (1953), 178–188. MR 52803, DOI
  • M. Krasnosselsky, Sur un critère pour qu’un domaine soit Ă©toilĂ©, Rec. Math. [Mat. Sbornik] N. S. 19(61) (1946), 309–310 (Russian, with French summary). MR 0020248
  • Steven R. Lay, Convex sets and their applications, John Wiley & Sons, Inc., New York, 1982. Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 655598
  • 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), no. 3, 281–293. Third ACM Symposium on Computational Geometry (Waterloo, Ont., 1987). MR 937288, DOI
  • Joseph O’Rourke, Art gallery theorems and algorithms, International Series of Monographs on Computer Science, The Clarendon Press, Oxford University Press, New York, 1987. MR 921437
  • 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.
  • Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264

Similar Articles

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

Retrieve articles in all journals with MSC: 52A30

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society