Let $F = \left\{ {({A_1} + {B_1})x + ({A_2} + {B_2}):\left\| {{B_i}} \right\| \le {\alpha _i},i = 1,2} \right\}$ such that, $A_i,B_i\in M_n$ and $\alpha_i>0$ for all $i$ and $x$ is a complex variable and ${\left\| . \right\|}$ is subordinate matrix norm.

What is boundary of $F$?

Can we say that boundary of $F$ equal $ \{ (A_1+B_1)x+(A_2+B_2):\|B_j\| \le \alpha_i$ for each $i $, and equality holds for at least one $i$ $\}$, or no?


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