Hostname: page-component-7bb8b95d7b-dtkg6 Total loading time: 0 Render date: 2024-09-12T08:35:32.195Z Has data issue: false hasContentIssue false
  • English
  • Français

On Plane Sections and Projections of Convex Sets

Published online by Cambridge University Press:  20 November 2018

H. Groemer
H. Groemer*
Affiliation:
The University of Arizona, Tucson, Arizona
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be a three-dimensional convex body. It has been conjectured (cf. 3) that one can always find a plane H such that the intersection K ∩ H is, in a certain sense, fairly circular. Instead of the plane section K ∩ H one can also consider the orthogonal projection of K onto H. Our aim in this paper is to prove some results concerning this type of problems. It appears that John has found similar theorems (cf. the remarks of Behrend, 1, p. 717). His proof of the first inequality of our Theorem 1 has been published (6). It is based on a property of the ellipse of inertia which will not be used in the present paper.

A non-empty compact convex set 5 which is contained in some plane of euclidean three-dimensional space E3 will be called a convex domain.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1331 - 1337
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Behrend, F., Uber einige Affininvarianten konvexer Bereiche, Math. Ann. 113 (1937), 713–747.Google Scholar
2. Behrend, F., Uber die kleinste umbeschriebene und die grbsste einbeschriebene Ellipse eines konvexen Bereichs, Math. Ann. 115 (1938), 379–411.Google Scholar
3. Croft, H. T., Research Problems, Mimeographed notes, Cambridge, 1967.Google Scholar
4. Danzer, L., Laugwitz, D., and Lenz, H., Uber das Lownersche Ellipsoid und sein Analogon unter den einem Eikôrper einbeschriebenen Ellipsoiden, Arch. Math. 8 (1957), 214–219.Google Scholar
5. Fenchel, W., Elementare Beweise und Anwendungen einiger Fixpunktsàtze, Mat. Tidsskr., part B, 3–4 (1932), 66–87.Google Scholar
6. John, F., An inequality for convex bodies, Univ. Kentucky Research Club Bull. (1940), no. 6, 26.Google Scholar
7. John, F., Extremum problems with inequalities as subsidiary conditions, Studies and essays presented to R. Courant, pp. 187–204 (Interscience, New York, 1948).Google Scholar
8. Leichtweiss, K., Uber die affine Exzentrizitat konvexer Kôrper, Arch. Math. 10 (1959), 187–199.Google Scholar
9. Zaguskin, V. L., On circumscribed and inscribed ellipsoids of extremal volume, Uspehi Mat. Nauk 13 (1958), 89–92.Google Scholar