Considering the space of closed subsets of $\mathbb{ R}^d$, endowed with the Chabauty-Fell topology, and the affine action of $SL_d(\mathbb{ R})\ltimes\mathbb{ R}^d$, we prove that the only minimal subsystems are the fixed points $\{ \varnothing\}$ and $\{ \mathbb{ R}^d\}$. As a consequence we resolve a question of Gowers concerning the existence of certain Danzer sets: there is no set $Y \subset \mathbb{ R}^d$ such that for every convex set $\mathcal{ C} \subset \mathbb{ R}^d$ of volume one, the cardinality of $\mathcal{ C} \cap Y$ is bounded above and below by nonzero contants independent of $\mathcal{ C}$. We also provide a short independent proof of this fact and deduce a quantitative consequence: for every $\varepsilon$-net $N$ for convex sets in $[0,1]^d$ there is a convex set of volume $\varepsilon$ containing at least $\Omega(\log\log(1/\varepsilon))$ points of $N$.
Categories: math.DS math.CO