Fekete's aperture-angle approximation conjecture

Let $C$ be a compact convex figure in the plane, let $\alpha(C,k)$ denote the best worst-case aperture-angle approximation of $C$ by an inscribed convex $k$-gon, and let $\alpha(k)=\inf_C\alpha(C,k)$. A regular $(k+1)$-gon has interior angle $\bigl(1-\frac{2}{k+1}\bigr)\pi$. Fekete's conjecture. For any $k\geq 2$, the smallest value of $\alpha(k)$ is achieved by the regular $(k+1)$-gon, and therefore

$$\alpha(k)=\bigl(1-\frac{2}{k+1}\bigr)\pi.$$

The conjecture was known for $k=2$ and $k=3$ and was supported by experiments; the paper proves it, first for polygons and then for arbitrary compact convex sets by a limit argument.