We define $$B_1(r,c) = \{x\in\mathbb{R}^d : \Vert{}x-c\Vert_1 \le r\}$$
Now for arbitrary constant $r \ge s > 0$, given constant $\epsilon \in (r-s,r+s)$, considering $v \in \mathbb{R}^d$ such that $\Vert{v}\Vert_1 = \epsilon$, what is the minimal volume of $B_1(r,0) \cap B_1(s,v)$ when $v$ varying?
Is there any idea or suggested reference?
I am sure that it goes to minimal when $v = (\epsilon, 0, ..., 0)$ after considering situations where $d = 2, 3$. But have no idea for strict proof. Any idea for proof?
Thank you very much.