Combinatorial problems on the illumination of convex bodies

Abstract

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.Mathematics Subject Classification (1991). 52A37, 52A40.

Convex solids whose point-source shadow-boundaries lie in hyperplanes

“…Due to various applications (see, e.g., Amir [1]), a characterization of solid ellipsoids as convex bodies with hyperplanar shadow-boundaries is one of the best known. The concept of shadow-boundary appears in various problems on illumination of convex sets by families of rays with a common direction or a common point source (see, e.g., [8][9][10]). We consider below illumination by rays with a common point source (see [18] for a similar problem involving illumination by parallel rays).…”

Symmetrization of Closure Operators and Visibility

“…On the other hand, we have studied in [11] visibility problems (see also the survey [10]) in terms of closure operators. Roughly speaking, a point lies in the closure of some set if, in a sense, it can be "seen" from this set.…”

Clouds of planar convex bodies

“…We close this paragraph with a challenging open problem due to Valeriu Soltan (see [4]) concerning the planar version of the problem. Is it true that 8 ≤ C T (K) ≤ 9 for every planar convex body K?…”

“…Suppose that there exist two concentric circles B and B with radius r = 0.99 and r = 1, respectively, such that B ⊆ K ⊆ B . Then C(K) = 8.For additional problems and results concerning clouds of different kinds we refer the reader to the excellent survey paper by Horst Martini and Valeriu Soltan[4].…”