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 $C$?



Polar interpretation of convexity
Convexity is equivalent to the function $r(\theta):[0,2\pi)\to\mathbb{R}^2$ being well-defined 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](http://en.wikipedia.org/wiki/Stewart%27s_theorem). (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 $1/2$).


