Schranken für die Dicke der Simplexe. (On bounds for the width of simplices) (Q582581)

Let T be a d-dimensional simplex that is circumscribed to a sphere of radius r. If \(\Delta\) denotes its width, i.e. its minimal breadth, and \(\bar r\) the radius of its circumsphere, then \textit{P. Steinhagen} [Abh. Math. Semin. Univ. Hamburg 1, 15-26 (1921)] has proved that \(\Delta \leq (1/d)c_ dr\), and \textit{R. Alexander} [Geom. Dedicata 6, 87-94 (1977; Zbl 0372.52003)] that \(\Delta \leq (1/d^ 2)c_ d\bar r\), where \(c_ d=2d\cdot (d+1)/\sqrt{d+2}\) if d is even, \(c_ d=2d\sqrt{d}\) if d is odd. The author gives a new, very nice and elementary proof of these inequalities which are sharp for regular T, by proving the theorem \[ \Delta \leq c_ d(V/S), \] where V, S denote the volume and the measure of the surface of T, respectively, Steinhagen's and Alexander's inequalities then follow immediately from \(V=(1/d)Sr\) and \(V\leq (1/d^ 2)S\bar r\).