Section: New Results

On Sets Avoiding Distance 1

In a joint work with C. Bachoc, T. Bellitto and P. Moustrou [11], we consider the maximum density of sets avoiding distance 1 in ${ℝ}^n$. Let $\left|\right|.\left|\right|$ be a norm of $R^n$ and $G_{\left|\right|.\left|\right|}$ be the so-called unit distance graph with the points of $R^n$ as vertex set and for edge set, the set of pairs $\{x,y\}$ such that $\left|\right|x-y\left|\right|=1$. An independent set of $G_{\left|\right|.\left|\right|}$ is said to avoid distance 1.

Let ${\left|\right|.\left|\right|}_E$ denote the Euclidean norm. For $n=2$, the chromatic number of $G_{{\left|\right|.\left|\right|}_E}$ is still wide open: it is only known that $4\le \chi \left(G_{{\left|\right|.\left|\right|}_E}\right)\le 7$ (Nelson, Isbell 1950). The measurable chromatic number ${\chi }_m$ of the graph $G_{\left|\right|.\left|\right|}$ is the minimal number of measurable stable sets of $G_{\left|\right|.\left|\right|}$ needed to cover all its vertices. Obviously, we have $\chi \left(G_{{\left|\right|.\left|\right|}_E}\right)\le {\chi }_m\left(G_{{\left|\right|.\left|\right|}_E}\right)$. For $n=2$, $5\le {\chi }_m\left(G_{{\left|\right|.\left|\right|}_E}\right)$ (Falconer 1981).

Let $m_1\left(G_{\left|\right|.\left|\right|}\right)$ denote the maximum density of a measurable set avoiding distance 1. We have $\frac{1}{m_1\left(G_{\left|\right|.\left|\right|}\right)}\le {\chi }_m\left(G_{\left|\right|.\left|\right|}\right)$. We study the maximum density $m_1$ for norms defined by polytopes: if $P$ is a centrally symmetric polytope and $x$ is a point of $R^n$, ${\left|\right|x\left|\right|}_P$ is the smallest positive real $t$ such that $x\in tP$. Polytope norms include some usual norms such as the $L^1$ and $L^{\infty }$ norms.

If $P$ tiles the space by translation, then it is easy to see that $m_1\left(G_{{\left|\right|.\left|\right|}_P}\right)\ge \frac{1}{2^n}$. C. Bachoc and S. Robins conjectured that equality always holds. We show that this conjecture is true for $n=2$ and for some polytopes in higher dimensions.