On circumscribed cylinders of convex sets
type: Item
instance of: scholarly article
title: On circumscribed cylinders of convex sets
zbMATH Open document ID: 0787.52005
DOI: 10.1007/BF01263623
published in: Geometriae Dedicata
publication date: 1993-08-17T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 205517
zbMATH Keywords: \(n\)-dimensional convex sets
zbMATH Keywords: area
zbMATH Keywords: circumscribed cylinders
zbMATH Keywords: inequalities
author: Helmut Groemer
reviewed by: Johannes Böhm
MaRDI profile type: Publication
cites work: Q3159163
cites work: Geometric Inequalities for Plane Convex Bodies
cites work: An Area-Width Inequality for Convex Curves
cites work: A note on a theorem of H. Knothe
cites work: Stability Results for First Order Projection Bodies
cites work: Q4274032
cites work: Q3245768
cites work: Inversion of two theorems of Archimedes
cites work: Mixed width-integrals of convex bodies
cites work: Gleitkörper in konvexen Polytopen.
cites work: The mean surface area of the boxes circumscribed about a convex body
cites work: \(L_p\) metrics for compact convex sets
full work available at URL: https://doi.org/10.1007/bf01263623
OpenAlex ID: W2095312152
description: scientific article; zbMATH DE number 205517