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 $\frac 1 2 \leq \frac 1 2 (0)+\frac 1 2 (0)$!
In $[0,1]$, $$f:=x-x^3,\\g:=(1-x)-(1-x)^3,\\h:=\min(f,g)$$

Now the average of $h\left(\dfrac1{\sqrt 3}\right)$ and $h\left(1-\dfrac1{\sqrt 3}\right)$ is $\dfrac{2}{3\sqrt3}$, but $h\left(\dfrac12\right)=\dfrac38$, which is smaller.