Positive centers and the Bonnesen inequality
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- by Michael E. Gage PDF
- Proc. Amer. Math. Soc. 110 (1990), 1041-1048 Request permission
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.References
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T. Bonnesen, Les problemes des isoperimetres et des isepiphanes, Gauthier-Villars et Cie, Paris, 1929.
- William J. Firey, Shapes of worn stones, Mathematika 21 (1974), 1–11. MR 362045, DOI 10.1112/S0025579300005714
- Michael E. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), no. 4, 1225–1229. MR 726325, DOI 10.1215/S0012-7094-83-05052-4
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 1041-1048
- MSC: Primary 52A40
- DOI: https://doi.org/10.1090/S0002-9939-1990-1042266-7
- MathSciNet review: 1042266