Homothetic covering and the illumination problem of convex bodies (2024)
At a first glance, the problem of illuminating the boundary of a convex body by light sources and the problem of covering a convex body by smaller homothetic copies seem different. But actually they both are incarnations of the same open problem in convex and discrete geometry, the illumination conjecture. This conjecture is also known as the Gohberg-Markus-Hadwiger conjecture, referring to some of its original proposers. In fact the two different versions of the problem were poses independently of one another and later shown that they were equivalent.\\ In 1957 Hadwiger posed the problem of finding the smallest natural number $N$ such that any $d$-dimensional convex body can be covered by the interior of a union of at the most $N$ of its translates. In the 1960 the problem was translated in terms of smaller homothetical copies of the original body. Later, in 1960, Boltyanski introduced the problem of illuminating the boundary of a convex body by the smallest amount of external light sources. For a given convex body we call the answer to both of the previous problems the illumination number of the body. It is conjectured that every $d$-dimensional convex body has an illumination number smaller than or equal to $2^d$ with equality for $d$-dimensional parallelepipeds. It turns, out that this is one of the central problems in convex and discrete geometry.\\ Our work consists of studying the advancements done to solve the conjecture in various possible approaches.