Geodesic diameter of bodies of constant width (Q844462)

Let \(\Phi\) be a three-dimensional body of constant width \(B\). The geodesic diameter \(G\) of the surface of \(\Phi\) is estimated via \(B\) from above and from below. The main result states that the inequality \(G\leq \frac{\pi}{2} B\) holds, moreover the equality \(G= \frac{\pi}{2} B\) occurs if and only if \(\Phi\) is a body of revolution. The question is what is the optimal estimate from below for \(G\) in terms of \(B\). What is an extremal body, which has the minimal geodesic diameter \(G\) between all the convex bodies of a fixed constant width \(B\)?