On levels of the distance function from the boundary of convex domain (Q1190708)

Let \(D\) be a 2-disc with a Riemannian metric \(g\) of nonnegative curvature and strictly convex boundary curve \(\partial D\). Let \(d^*(D)\) denote the farthest distance of points of \(D\) from \(\partial D\) and \(p\) the soul, i.e. the (unique) point where \(d^*(D)\) is achieved. Under the assumption that \((D,g)\) is real holomorphic, the author deduces an upper bound for the quantity area \((D)/(d^*(D))^ 2\). The bound is \(\pi\) if there are infinitely many minimizing geodesics from the soul \(p\) to \(\partial D\), and otherwise depends on the angles between these geodesics at \(p\).