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.

When? | 07.11.2017 17:15 |
---|---|

Where? | PER 08 Phys 2.52 Chemin du Musée 3 1700 Fribourg |

Contact | Department of Mathematics isabella.schmutz@unifr.ch |