On circumscribed cylinders of convex sets
In 1957 \textit{H. Knothe} [Mich. Math. J. 4, 53-56 (1957; Zbl 0077.359)] had shown an inequality which enabled him to prove the following converse of a theorem of Archimedes: If for every right circumscribed cylinder of a given three-dimensional convex body \(K\) the surface area of the lateral portion of the boundary of this cylinder equals the surface area of \(K\), then \(K\) is a ball. -- Two years later \textit{Wm. Firey} presented another simple proof [Mich. Math. J. 6, 53-54 (1959; Zbl 0085.167)]. In this note two inequalities are given concerning the average size of circumscribed cylinders of convex sets. These inequalities generalize and supplement Knothe's result in the following way: There is no restriction on the dimension, the inequality for the mean value of the size of circumscribed cylinders is strengthened so that it also yields a stability estimate, and a tight upper bound for this mean value is established. All the cases are discussed where equality holds. The converse of the above theorem of Archimedes follows from one of these inequalities for dimension 3. -- The proof uses \(d\)-dimensional spherical harmonics and furthermore results on mean projection measures (Quermassintegrale).
scientific article; zbMATH DE number 205517