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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Positive centers and the Bonnesen inequality

Author: Michael E. Gage
Journal: Proc. Amer. Math. Soc. 110 (1990), 1041-1048
MSC: Primary 52A40
MathSciNet review: 1042266
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Abstract: A positive center of a convex curve is a point from which the function $ p(\theta )L - A - \pi p{(\theta )^2}$ is positive for all values of $ \theta $. The support function is $ p$ and $ L$ and $ A$ are the length and area of the curve, respectively. This paper proves that all convex curves have a positive center and gives an example which shows that the common centroids (Steiner point, etc.) are not necessarily positive centers. A strengthened version of the Bonnesen inequality is obtained and a simplified proof of the one-dimensional Firey inequality.

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Keywords: Bonnesen inequality, symmetrization
Article copyright: © Copyright 1990 American Mathematical Society

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