On the Density of Sets Avoiding Parallelohedron Distance 1

Let $\Vert \cdot \Vert$ be a norm on $\mathbb{R}^n$. We consider the so-called unit distance graph $G$ associated with $\Vert \cdot \Vert$: the vertices of $G$ are the points of $\mathbb{R}^n$, and the edges connect the pairs $\{x,y\}$ satisfying $\Vert x-y\Vert=1$. We define $m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)$ as the supremum of the densities achieved by independent sets of $G$. The number $m_1$ was introduced by Larman and Rogers (1972) as a tool to study the measurable chromatic number $\chi_m(\mathbb{R}^n)$ of $\mathbb{R}^n$ for the Euclidean norm.