What is the supremum density of a measurable set in $\mathbb R^n$ avoiding distance 1?
If the distance is the Euclidean distance, the answer is known only in
the trivial case n = 1. This problem is closely related to that of the
determination of the chromatic number of the Euclidean space, a
surprisingly difficult problem even in dimension 2, which is open
since it was posed by Nelson and Hadwiger in 1950. We will discuss
recent results obtained on this question and also on several variants
that rely on a combination of methods from convex optimization and
Fourier analysis.
In one of these variants, one replaces the Euclidean norm by a norm defined
by a convex symmetric polytope. When the polytope tiles the space by
translations, we conjecture that the answer is $\mathbb 1/2^n$ and that the
proof should be much easier that in the Euclidean setting.
We will present a proof of this conjecture in dimension 2 and discuss
a few other cases of Voronoi polytopes associated to lattices.