Inequalities between the radii of some spheres connected with a convex surface (Q1288105)

Let \(\Phi\subset\mathbb R^n\), \(n\geq 1\), be a bounded convex body with nonempty interior. Let \(\lambda\) be the radius of the biggest sphere which can be rolled on the inward side of \(\Phi\), let \(\Lambda\) be the radius of the inscribed sphere for \(\Phi\), let \(M \) be the radius of the circumscribed sphere for \(\Phi\), and let \(\mu\) be the radius of the smallest sphere such that \(\Phi\) can be rolled on its inward side. Obviously, the numbers \(\lambda\), \(\Lambda\), \(M \), and \(\mu\) satisfy either the conditions \[ 0<\lambda < \Lambda <M <\mu \tag{1} \] or the conditions \[ 0<\lambda = \Lambda =M =\mu. \tag{2} \] The author proves that, given a quadruple \((\lambda, \Lambda, M , \mu)\) satisfying either (1) or (2), there exists a convex bounded surface \(\Phi\) with nonempty interior if and only if the inequality \[ (M -\lambda)^2 +(\mu - \Lambda)^2\leq (\mu - \lambda)^2 \] holds. The author deduces the above assertion from a more general theorem (which is also proven in the article under review, but whose statement is rather too complicated to be given here) and shows that this more general theorem also implies the following generalization of the Bonnet theorem: If the Gauss curvature of a convex surface \(\Phi\subset\mathbb R^n\), \(n\geq 3\), is greater than or equal to 1 at every point, then there exists an open ball of radius \(\pi/2\) which contains \(\Phi\); moreover, the number \(\pi/2\) cannot be replaced by any smaller number. A slightly weakened version of the last statement may be found in the book by \textit{W. Blaschke} [`Kreis und Kugel' (Reprint, Chelsea, New York) (1949; Zbl 0041.08802)].