On the illumination problem for convex sets (Lenny Fukshansky, CMC)

Let K be a compact convex set in the Euclidean space R^n. How many lights are needed to illuminate its boundary? A classical conjecture of Boltyanskii (1960) asserts that 2^n lights are sufficient to illuminate any such set K. While this is still open, an earlier observation of Hadwiger (1945) guarantees that if K has smooth boundary, then n+1 lights are sufficient: we only need to position these lights at the vertices of a simplex containing K in its interior. In fact, this observation allows us to estimate how far from K these lights need to be. A more delicate problem arises if we insist on placing the lights at points of a fixed lattice L: how far from K must the lights be then? We discuss this problem, producing a bound on this distance, which depends on certain orthogonality and symmetry properties of the lattice in question. Interestingly, for some nice classes of lattices, a bound independent of L can be produced.