Optimality of lower bound for ε-nets/dispersion and implications for Danzer’s problem

Establish whether the lower bound obtained by Rote and Tichy (1996) for the equivalent combinatorial problem concerning ε-nets in the range space of boxes in a cube is optimal; if it is optimal, this would imply a negative answer to Danzer’s problem (the existence of a finite-density set in R^d intersecting every convex body of fixed volume).

Background

Danzer’s problem asks whether there exists a set of finite density in R^d intersecting every convex set of a fixed volume. Work by Solomon and Weiss reduced aspects of this to an equivalent combinatorial problem involving ε-nets for the range space of boxes in a cube, connecting Danzer-type questions with dispersion and ε-net theory.

Rote and Tichy (1996) studied dispersion and related lower bounds. The paper cites a conjecture that the lower bound for the equivalent ε-net/dispersion problem is best possible, which would in turn imply a negative solution to Danzer’s problem.