We define $B(c,r) = \{x \in \mathbb{R} \ : \ |x - c| \le r\}$ with $r \ge 0$. Given two balls $b_1 = B(c_1, r_1)$ and $b_2 = B(c_2, r_2)$, multiplication is defined as follows: $$ b_1 * b_2 = B(c_1 c_2, r_1 r_2 + |c_1| r_2 + |c_2|r_1). $$ I've taken this equation from (7) in this thesis. An alternative formula is given in (16) but it simply involves replacing $r_1 r_2$ with $2 r_1 r_2$.
I'm trying to prove that if $x \in b_1$ and $y \in b_2$ then $xy \in b_1 * b_2$, which certainly needs to be true for ball multiplication to make sense.
An issue is that we could have $x$ positive but $c_1$ negative so if we were to split by cases we would need to consider nonnegative and negative cases for $x$, $y$, $c_1$ and $c_2$ which gives us a total of 16 cases. Even then, I'm not completely sure that the proof would go through.
It seems like there should be a more straightforward proof so I was wondering if I'm missing some trick here.
