aequationes mathematicae

, Volume 57, Issue 2–3, pp 121–152 | Cite as

Combinatorial problems on the illumination of convex bodies

  • H. Martini
  • V. Soltan


This is a review of various problems and results on the illumination of convex bodies in the spirit of combinatorial geometry. The topics under review are: history of the Gohberg-Markus-Hadwiger problem on the minimum number of exterior sources illuminating a convex body, including the discussion of its equivalent forms like the minimum number of homothetic copies covering the body; generalization of this problem for the case of unbounded convex bodies; visibility and inner illumination of convex bodies; primitive illuminating systems for convex bodies; illumination and visibility of families of convex bodies; clouds formed by translates or homothetic copies of a convex body; miscellaneous results.

Keywords. Belt body, belt polytope, body of constant width, centrally symmetric convex body, cloud, convex body, covering by homothetic copies, cyclic polytope, exterior illumination by points or directions, Gohberg-Markus-Hadwiger problem, family of convex bodies, inner illumination, isothetic rectangles, moment curve, nearly conic body, parallelotope, primitive illuminating system, shadow-boundary, unbounded convex body, visibility by points or directions, zonotope, zonoid. 


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Copyright information

© Birkhäuser Verlag, Basel, 1999

Authors and Affiliations

  • H. Martini
    • 1
  • V. Soltan
    • 2
  1. 1.Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, GermanyDE
  2. 2.Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030, USAUS

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