Let $P$ be a convex polygon with perimeter $N$, and let $x$ be an arbitrary point. Show that the maximum distance between $x$ and a vertex of $P$ is at least proportional to $N$ (i.e. there exists a constant $\alpha$ and a vertex $V$ of the polygon such that $dist(x,V) \geq \alpha N$).
The worst case seems to be that of regular polygons and $x$ the centroid, which suggests that $\alpha$ should be smaller than $\frac{1}{2\pi}$.
This looks like there is an elementary argument, but it is eluding me...