Abstract: Let us think of a convex body in R^n (convex, compact set, with non-empty interior) as an opaque object, and let us place point light sources around it, wherever and however far from the body we want, to illuminate its entire surface. What is the minimum number of such light sources that we would have to use? The Hadwiger-Boltyanski illumination conjecture from 1960 states that we need at most as many light sources as for the n-dimensional hypercube, and more generally, as for n-dimensional parallelotopes. For the latter their illumination number is exactly 2^n, and they are conjectured to be the only equality cases.

The conjecture is still open in dimension 3 and above, and has only been fully settled for certain classes of convex bodies (e.g. zonoids, bodies of constant width, etc.). In this talk I will briefly discuss some of its history, and then focus on joint works with Wen Rui Sun that settle the conjecture for all 1-symmetric convex bodies (by complementing a method previously developed by K. Tikhomirov for this class of bodies), and that also deal with certain cases of 1-unconditional bodies, and possible extensions of those.