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Geometric Inequalities for Plane Convex Bodies

Published online by Cambridge University Press:  20 November 2018

G. D. Chakerian
G. D. Chakerian*
Affiliation:
Department of Mathematics, University of California, Davis, Ca. 95616
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In what follows we shall mean by a plane convex body K a compact convex subset of the Euclidean plane having nonempty interior. We shall denote by h (K, θ) the supporting function of K restricted to the unit circle. This measures the signed distances from the origin to the supporting line of K with outward normal (cos θ, sin θ). The right hand and left hand derivatives of h (K, θ) exist everywhere and are equal except on a countable set.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 22 , Issue 1 , 01 March 1979 , pp. 9 - 16
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Blaschke, W., Beweise zu Sätzen von Brunn und Minkowski über die Minimaleigenschaft des Kreises, Jber. dtsch. Math.-Ver. 23 (1914), 210-234.Google Scholar
2. Bol, G., Beweis einer Vermutung von H. Minkowski, Abh. math. Sem. Univ. Hamburg 15 (1943), 37-56.Google Scholar
3. Bonnesen, T. and Fenchel, W., Theorie der konvexen Körper, Berlin, 1934.Google Scholar
4. Chakerian, G. D., The mean volume of boxes and cylinders circumscribed about a convex body, Israel J. Math. 12 (1972), 249-256.Google Scholar
5. Chakerian, G. D., Isoperimetric inequalities for the mean width of a convex body, Geom. Dedicata 1 (1973), 356-362.Google Scholar
6. Chakerian, G. D., and Sangwine-Yager, J. R., A generalization of Minkowski's inequality for plane convex sets, to be published.Google Scholar
7. Chernoff, P. R., An area-width inequality for convex curves, Amer. Math. Monthly 76 (1969), 34-35.Google Scholar
8. Fenchel, W. and Jessen, B., Mengenfunktionen und konvexe Körper, Det Kgl. Danske Videnskabernes Selskab. Mat.-fys. Meddelelser 16, no. 3, 1938.Google Scholar
9. Hadwiger, H., Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Berlin-Göttingen- Heidelberg, 1957.Google Scholar
10. Heil, E., Eine Verschärfung der Bieberbachschen Ungleichung und einige andere Abschätzungen für ebene konvexe Bereiche, Elem. Math. 27 (1972), 4-8.Google Scholar
11. Hurwitz, A., Sur quelques applications géométriques des séries de Fourier, Annales de l'École Normale supérieure, Ser. 3, 19 (1902), 357-408.Google Scholar
12. Lutwak, E., Mixed width-integrals of convex bodies, Israel J. Math. 28 (1977), 249-253.Google Scholar
13. Radziszewski, K., Sur une fonctionelle défine sur les ovales, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, 10 (1956), 57-59.Google Scholar
14. Schneider, R., The mean surface area of the boxes circumscribed about a convex body, Ann. Polon. Math. 25 (1972), 325-328.Google Scholar