Section: New Results
On sets avoiding distance 1
In a joint work with C. Bachoc, T. Bellitto and P. Moustrou [39], we consider the maximum density of sets avoiding distance 1 in . Let be a norm of and be the so-called unit distance graph with the points of as vertex set and for edge set, the set of pairs such that . An independent set of is said to avoid distance 1.
Let denote the Euclidean norm. For , the chromatic number of is still wide open: it is only known that (Nelson, Isbell 1950). The measurable chromatic number of the graph is the minimal number of measurable stable sets of needed to cover all its vertices. Obviously, we have . For , (Falconer 1981).
Let denote the maximum density of a measurable set avoiding distance 1. We have . We study the maximum density for norms defined by polytopes: if is a centrally symmetric polytope and is a point of , is the smallest positive real such that . Polytope norms include some usual norms such as the and norms.
If tiles the space by translation, then it is easy to see that . C. Bachoc and S. Robins conjectured that equality always holds. We show that this conjecture is true for and for some polytopes in higher dimensions.