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A Class of Star-Shaped Bodies

Published online by Cambridge University Press:  20 November 2018

Z.A. Melzak
Z.A. Melzak*
Affiliation:
McGill University
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The more important properties of the class κ of all bounded convex bodies in E3 with non-empty interior include: uniform approximability by polyhedra, existence of volume and surface area, and Blaschke's selection principle, [l ], [2 ]. In this note we define and consider a class ℋ of star-shaped bodies in E3, which enjoys many properties of κ, among them the above-mentioned ones, and is considerably larger. Roughly speaking, ℋ consists of closed bounded sets in E3 with nonempty interior, whose boundary is completely visible from every point of a set with non-empty interior. It turns out that ℋ is identifiable with the class of all real-valued positive functions on the sphere S3 which satisfy a Lipschitz condition.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 2 , Issue 3 , 01 September 1959 , pp. 175 - 180
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Bonnesen, T. and Fenchel, W., Konvexe Koerper, (Chelsea, 1948).Google Scholar
2. Blaschke, W., Kreis und Kugel, (Chelsea, 1949).Google Scholar
3. Rademacher, H., Partielle and totale Differenzierbarkeit, Math. Annalen 81 (1920), 52.Google Scholar
4. Rado, T., Flaechenmass rectifizierbarer Flaechen, Math. Annalen 100 (1928), 445.Google Scholar