On problems of Danzer and Gowers and dynamics on the space of closed subsets of ${\mathbb{R}}^d$

Considering the space of closed subsets of ${\mathbb{R}}^d$, endowed with the Chabauty-Fell topology, and the affine action of $S{L}_d(\mathbb{R})⋉{\mathbb{R}}^d$, we prove that the only minimal subsystems are the fixed points $\{\mathrm{\varnothing }\}$ 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 $\epsilon$-net $N$ for convex sets in $[0,1{]}^d$ there is a convex set of volume $\epsilon$ containing at least $\mathrm{\Omega }(\log \log (1\mathrm{/}\epsilon ))$ points of $N$.