Danzer set
Ludwig Danzer asked whether it is possible for such a set to have bounded density.
[3] A Danzer set, in an n-dimensional Euclidean space, is a set of points in the space that has a non-empty intersection with every convex body whose n-dimensional volume is one.
[4] Danzer's question asked whether, more strongly, the average number of points per unit area could be bounded.
[1] One way to define the problem more formally is to consider the growth rate of a set
-dimensional Euclidean space, defined as the function that maps a real number
If so, this would equal the growth rate of well-spaced point sets like the integer lattice (which is not a Danzer set).
[1] An equivalent formulation involves the density of a set
Danzer's question asks whether there exists a Danzer set of bounded density or, alternatively, whether every set of bounded density has arbitrarily high-volume convex sets disjoint from it.
[3] Instead of asking for a set of bounded density that intersects arbitrary convex sets of unit volume, it is equivalent to ask for a set of bounded density that intersects all ellipsoids of unit volume, or all hyperrectangles of unit volume.
For instance, in the plane, the shapes of these intersecting sets can be restricted to ellipses, or to rectangles.
[3] It is possible to construct a Danzer set of growth rate that is within a polylogarithmic factor of
For instance, overlaying rectangular grids whose cells have constant volume but differing aspect ratios can achieve a growth rate of
[5] A construction for Danzer sets is known with a somewhat slower growth rate,
[4] Because both the overlaid grids and the improved construction have growth rates faster than
, these sets do not have bounded density, and the answer to Danzer's question remains unknown.
[3][4] Although the existence of a Danzer set of bounded density remains open, it is possible to restrict the classes of point sets that may be Danzer sets in other ways than by their densities, ruling out certain types of solution to Danzer's question.
[4] A strengthened variation of the problem, posed by Timothy Gowers, asks whether there exists a Danzer set
[6] This version has been solved: it is impossible for a Danzer set with this property to exist.
John Horton Conway recalled that, as a child, he slept in a room with wallpaper whose flower pattern resembled an array of dead flies, and that he would try to find convex regions that did not have a dead fly in them.
[8] In Conway's formulation, the question is whether there exists a Danzer set in which the points of the set (the dead flies) are separated at a bounded distance from each other.
Such a set would necessarily also have an upper bound on the distance from each point of the plane to a dead fly (in order to touch all circles of unit area), so it would form a Delone set, a set with both lower and upper bounds on the spacing of the points.
, so if it exists then it would also solve the original version of Danzer's problem.
Conway offered a $1000 prize for a solution to his problem,[8][9] as part of a set of problems also including Conway's 99-graph problem, the analysis of sylver coinage, and the thrackle conjecture.
