Positive centers and the Bonnesen inequality
Michael E. Gage
Proc. Amer. Math. Soc. 110 (1990), 1041-1048
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Abstract: A positive center of a convex curve is a point from which the function is positive for all values of . The support function is and and 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.
T. Bonnesen, Les problemes des isoperimetres et des isepiphanes, Gauthier-Villars et Cie, Paris, 1929.
J. Firey, Shapes of worn stones, Mathematika
21 (1974), 1–11. MR 0362045
E. Gage, An isoperimetric inequality with applications to curve
shortening, Duke Math. J. 50 (1983), no. 4,
1225–1229. MR 726325
- T. Bonnesen, Les problemes des isoperimetres et des isepiphanes, Gauthier-Villars et Cie, Paris, 1929.
- W. Firey, Shapes of worn stones, Mathematica 21 (1974), 1-11. MR 0362045 (50:14487)
- M. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), 1225-1229. MR 726325 (85d:52007)
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