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Minkowski surface area under affine transformations

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Rolf Clack
Rolf Clack
Affiliation:
Mathematisches Institut der Albert-Ludwigs-Universität, Hebelstrasse 29, 7800 Freiburg im Br., Germany.
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Extract

A convex compact subset of ℝd is called a convex body. The (Euclidean) surface area and volume of a convex body K are denoted s(K) and v(K) respectively. The support function of a convex body K is denned by h(K, x) = maxy∈K xty and the polar dual of K is given by K0 = {x: |xty|1, y∈K}. Double vertical bars shall denote the Euclidean length of a vector , and S shall denote the unit sphere (the Euclidean unit ball): S = {x: ║x║≤1}. We use for the mixed volume

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 37 , Issue 2 , December 1990 , pp. 232 - 238
Copyright
Copyright © University College London 1990

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References

1. Benson, R. V.. Euclidean Geometry and Convexity (McGraw-Hill, 1966).Google Scholar
2. Bonnesen, T. and Fenchel, W.. Theory of Convex Bodies. Translated by Boron, , Christenson, and Smith, (BCS Associates, Idaho, USA, 1987). Original German edition copyright 1934 by Springer, Berlin.Google Scholar
3. Busemann, H.. The isoperimetric problem for Minkowski area. Amer. J. Math., 71 (1949), 743–762.CrossRefGoogle Scholar
4. Busemann, H.. The foundations of Minkowski geometry. Comm. Math. Helvetici, 24 (1950), 156–187.CrossRefGoogle Scholar
5. Busemann, H.. A theorem on convex bodies of the Brünn-Minkowski type. Proc. Nat. Acad. Sci. U.S.A., 35 (1949), 27–31.CrossRefGoogle ScholarPubMed
6. Holmes, R. D. and Thompson, A. C.. N-dimensional area and content in Minkowski spaces. Pac. J. Math., 85 (1979), 77–110.CrossRefGoogle Scholar
7. Johnson, K. and Thompson, A. C.. On the isoperimetric mapping in Minkowski spaces. Colloquia Math. Soc. Jànos Bolyai, 48. Intuitive Geometry (Siófok, 1985), 273–287.Google Scholar
8. Lutwak, E.. Intersection bodies and dual mixed volumes. Adv. in Math., 71 (1988), 232–261.CrossRefGoogle Scholar
9. Petty, C. M.. Surface area of a convex body under affine transformations. Proc. Amer. Math. Soc., 12 (1961), 824–828.CrossRefGoogle Scholar
10. Thompson, A. C.. An equiperimetric property of Minkowski circles. Bull. London Math. Soc., 7 (1975), 271–272.CrossRefGoogle Scholar