Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or necessary for) convexity of $C$?

Convexity is equivalent to the function $r(\theta):[0,2\pi)\to\mathbb{R}^2$ being welldefined and satisfying the condition $$ r(\lambda \theta _1+(1\lambda)\theta _2) \geq \left \lambda r(\theta _1)+(1\lambda)r(\theta _2)\right$$ for all $\theta_1,\theta_2$ and $\lambda\in (0,1)$. I'm using vector valued functions here, but you can switch to ordinary functions easily by using Stewart's relation. (This is just rewriting that a segment whose endpoints are in the set must be in the set.) Of course since you are dealing with a polygon you only need to check $n$ inequalities (where $n$ is the number of sides). Namely pick $\theta_i$ to be the angle pointing towards the midpoint of the $i$th side, and pick arbitrary $\lambda$'s (say all equal to $1/2$). 

