In Boyd's text this is the converse to his example 3.5 and exercise 3.29 - I'm having a hard time convincing myself that the claim is true. As an example, consider $\mathbb{R}$ and the function $f$ defined to be:
\begin{cases} -x+1 & x \leq 1 \\ 2x-2 & 1 \leq x \leq 2 \\ -4x + 10 & 2 \leq x \end{cases}
Here is a plot of the three functions: Plot The idea behind their construction is for the third function $-4x + 10$ to dominate the first one and prevent the pointwise max from being the desired zig-zag (the pointwise max here looks like an $|x|$ with apex at $(2,2)$
What am I doing wrong?
Update: As pointed out below - the reason this counter-example works is because the claim only holds for convex piece-wise linear functions. In that case, this example would not be valid and we can prove it by leveraging the implied convexity of the functions epigraph, where any construct as above can be shown to violate the epigraphs convexity.