Denote $B_n\subset\Bbb R^n$ to be unit ball at origin.
Denote $S\subset B_n$ to region of type $\mathsf I$ if it satisfies $$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$
I am convinced $\mu(S)=0$ and likely is just a discrete set.
Denote $R\subset B_n$ to region of type $\mathsf{II}$ if it satisfies $$s\in R\iff\forall t\in R, s+t\in mB_n\mbox{ or }s-t\in mB_n$$
where $mB_n$ is unit ball scaled $m$ times where $m\in\Bbb R$.
Minimum $m$ needed so that $R=B_n$ is $\sqrt{2}$.
Among all such regions of type $\mathsf{II}$, what is $R$ with maximum volume with general $m\in[1,\sqrt{2}]$?
How many disjoint $R$ do you need to cover $p\%$ of $B_n$ when $m\in[1,\sqrt{2}]$?
Is there a simple geometric description such sets $R$ at least for those with maximum volume?