The Danzer problem and a solution to a related problem of Gowers

Is there a point set $Y$ in $R^d$, and $C>0$, such that every convex set of volume 1 contains at least one point of $Y$ and at most $C$? This discrete geometry problem was posed by Gowers in 2000, and it is a special case of an open problem posed by Danzer in 1965. I will present two proofs that answers Gowers’ question with a NO. The first approach is dynamical; we introduce a dynamical system and classify its minimal subsystems. This classification in particular yields the negative answer to Gowers’ question. The second proof is direct and it has nice applications in combinatorics. [This is a joint work with Omri Solan and Barak Weiss].