Combinatorial problems on the illumination of convex bodies
This is a comprehensive survey article on the well known Hadwiger's covering problem and on miscellaneous variants of the equivalent illumination problem. In particular, the most interesting related open questions and conjectures are recalled. The variety of subjects can be immediately seen from the titles of the sections: Gohberg-Marcus-Hadwiger problem, Generalizations for unbounded convex sets, Illumination of special convex bodies, Visibility of a convex body, Inner illumination of a convex body, Primitive illuminating systems for a convex body, Illumination and visibility of a family of convex bodies, Clouds, Miscellaneous results. The reader may compare the present survey with four other extensive surveys about Hadwiger's and illumination problems published in the decade 1990-1999: \textit{K. Bezdek} [New Trends in Discrete and Computational Geometry, Springer, 199-233 (1993; Zbl 0798.52004)], \textit{V. Boltyanskij} and \textit{I. Gohberg} [Nieuw Arch. Wisk. 13, 1-26 (1995; Zbl 0896.52023)], Chapter VI of the book \textit{V. Boltyanskij}, \textit{H. Martini}, and \textit{P. S. Soltan} [Excursions into Combinatorial Geometry, Springer (1997; Zbl 0877.52001)], and \textit{L. Szabó} [Bolyai Soc. Math. Stud. 6, 207-221 (1997; Zbl 0886.52005)].
scientific article; zbMATH DE number 1320493