Confusion regarding worst case robust optimization

Question

In a book, for an uncertain matrix $A(o)=A_0+o \cdot A_1$; where $o \in R$ is an uncertain scalar between a compact interval, then $\max_{-c\leq o \leq c} ||A(o)x-b||$

is re-written as maximum of,

$\{||(A_0-o.A_1)x||,||(A_0+o.A_1)x||\}$

Answer 1

Fact. A convex function on an interval attains its maximum at an extremity of the interval.

This is easy to proof, but to save space an focus on real issues, here is a like to a simple proof https://qr.ae/pNrZo6.

Now, for any $x \in \mathbb R^n$, the given function is convex on the interval $[-c, c]$ , and so in view of my introductory comment, you have $\max_{o \in [-c,c]}f(o) = \max(f(-c),f(c))$. Conclude.