Question: My understanding is that taking the minimum of two (or more) functions is like creating a union of the functions which in some cases would result in a non-convex function. I can draw it out with some simple 2d functions but I can't figure out how to show it with equations. Thanks in advance for the help!

Answer: 
Minimum of $|x|$ and $|1-x|$ is not convex. (In fact it is strictly concave on $(0,1)$). 
Write $\frac 1 2$ as $\frac 1 2 (0)+\frac 1 2 (1)$. Convexity would mean $\frac 1 2 \leq \frac 1 2 (0)+\frac 1 2 (0)$!
The minimum $h:\Bbb R \to \Bbb R$ of $f(x)=(x-1)^2$ and $g(x)=(x+1)^2$ is not convex at $[-1,1]$.

*Proof*: $h(0)>h(-1)$ and $h(0)>h(1)$.