The Geometry of Homothetic Covering and Illumination

  • Károly Bezdek
  • Muhammad A. Khan
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 234)


At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two sides of the same coin and give rise to one of the important longstanding open problems in discrete geometry, namely, the Illumination Conjecture. In this paper, we survey the activity in the areas of discrete geometry, computational geometry and geometric analysis motivated by this conjecture. Special care is taken to include the recent advances that are not covered by the existing surveys. We also include some of our recent results related to these problems and describe two new approaches – one conventional and the other computer-assisted – to make progress on the illumination problem. Some open problems and conjectures are also presented.


Illumination number Illumination conjecture Covering conjecture Separation conjecture X-ray number X-ray conjecture Illumination parameter Covering parameter Covering index Cylindrical covering parameters \(\epsilon \)-net of convex bodies 

MSC (2010):

52A37 52A40 52C15 52C17 



The first author is partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. The second author is supported by a Vanier Canada Graduate Scholarship (NSERC) and Alberta Innovates Technology Futures (AITF).


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada
  2. 2.Department of MathematicsUniversity of PannoniaVeszprémHungary

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