2 problems

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$ whose unit ball tiles $\mathbb{R}^n$ by translation. Let $m_1(\mathbb{R}^n,\|\cdot\|)$ denote the supremum of the upper densities of mea…

In the normed vector space $(\mathbb{R}^n,\|\cdot\|)$, a measurable set $A\subseteq\mathbb{R}^n$ avoids distance 1 if $\|x-y\|\neq 1$ for all $x,y\in A$. Let…