Brass's subpolygon Hausdorff-approximation conjecture

Let $\mathcal{P}$ be a family of convex polygons in $\mathbb{R}^{2}$ that is closed under taking subpolygons, where a subpolygon is the convex hull of a subset of the polygon's vertices. Say that an element of $\mathcal{P}$ is hardest to approximate by its $k$-vertex subpolygons with respect to the Hausdorff metric if it has the worst such approximation error. Brass's conjecture. If $\mathcal{P}$ has an element that is hardest to approximate by its $k$-vertex subpolygons with respect to the Hausdorff metric, then one can also find a $(k+1)$-gon in $\mathcal{P}$ with this property. The paper proves the conjecture by showing that every worst-approximable polygon under Hausdorff approximation is a $(k+1)$-gon.