On the exact constant in the quantitative Steinitz theorem in the plane (Q1338465)

A well-known theorem of Caratheodory states that any point in the convex hull of a point set \(P \subseteq R^ d\) is in the convex hull of some \(d + 1\) points of \(P\). Similarly for balls, a theorem of Steinitz states that there is a constant \(c(d) < 1\) such that if a ball of radius \(r\) entirely contained in the convex hull of \(P\), the concentric ball of radius \(c(d) r\) is entirely contained in the convex hull of some \(2d\) points of \(P\). For the plane \((d = 2)\), the authors determine the exact maximum value of the constant \(c(d) = c(2) = {\cos (2 \pi/5) \over \cos (\pi/5)} = {3 - \sqrt 5 \over 2} \geq 0.381966\). This means that, for any point set whose convex hull contains the unit ball, a subset of four points can always be found such that its convex hull contains the ball of radius \(c(2)\). The constant is tight in the case of the regular pentagon. In addition, they show that if 4 is replaced by 5, the exact constant increases to \(c_ 5 = {\cos (2 \pi/6) \over \cos (\Pi/6)} = 1/ \sqrt 3\), and more generally, provide some evidence that if 4 is replaced by \(k\), then \(c_ k\) equals exactly \({\cos (2 \pi/(k + 1)) \over \cos (\pi/(k + 1))}\) (it is always more).