On Problems of Danzer and Gowers and Dynamics on the Space of Closed Subsets of R^d

Considering the space of closed subsets ofRd, endowed with the Chabauty-Fell topology, and the affine action of SLd(R)Rd, we prove that the only minimal subsystems are the fixed points and {Rd}. As a consequence we resolve a question of Gowers concerning the existence of certain Danzer sets: there is no set Y Rd such that for every convex set C Rd of volume one, the cardinality of C n Y is bounded above and below by non-zero constants independent of C. We also provide a short independent proof of this fact and deduce a quantitative consequence: for every e-net N for convex sets in [0, 1]d there is a convex set of volume e containing at least (log log(1/e)) points of N.