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A Compactness Theorem For AffineEquivalence-Classes of Convex Regions

Published online by Cambridge University Press:  20 November 2018

A. M. Macbeath
A. M. Macbeath*
Affiliation:
Princeton University
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In some parts of the Geometry of Numbers it is convenient to know that certain affine invariants associated with convex regions attain their upper and lower bounds. A classical example is the quotient of the critical determinant by the content (if the region is symmetrical) for which Minkowski determined the exact lower bound 2–n. The object of this paper is to prove that for continuous functions of bounded regions the bounds are attained. The result is, of course, deduced from the selection theorem of Blaschke, and itself is a compactness theorem about the space of affine equivalence-classes.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 3 , 1951 , pp. 54 - 61
Copyright
Copyright © Canadian Mathematical Society 1951

References

[1] Bonnesen, T. and Fenchel, W., Théorie der konvexen Kö rper (Ergebnisse der Math.) (Berlin, 1943).Google Scholar
[2] Blaschke, W., Kreis und Kugel (Leipzig, 1916).Google Scholar
[3] Mahler, K., Ein Übertragungsprinzip fur konvexe Körper Casopis pro Pestovani Mat. a Fys., vol. 68 (1939), 93–102.Google Scholar