Comptes Rendus
no. 2
Geometry
The ε-positive center set and its applications

Presented by: Étienne Ghys

Shengliang Pan1; Yunlong Yang1; Pingliang Huang2
1 Mathematics Department, Tongji University, Shanghai, 200092, PR China
2 Mathematics Department, Shanghai University, Shanghai, 200444, PR China
Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 195-200.

In this paper we will first give a positive answer to Kaiser's conjecture on ε-positive centers for convex curves and then present its two applications.

Dans cette Note, nous apportons une réponse positive à la conjecture de Kaiser sur les centres ε-positifs des courbes convexes, puis nous en présentons deux applications.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.10.021
Keywords: Constant width curve, ε-Positive center set, Inner parallel body, Kaiser's conjecture, Positive center set

Shengliang Pan 1; Yunlong Yang 1; Pingliang Huang 2

1 Mathematics Department, Tongji University, Shanghai, 200092, PR China
2 Mathematics Department, Shanghai University, Shanghai, 200444, PR China 
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Shengliang Pan; Yunlong Yang; Pingliang Huang. The ε-positive center set and its applications. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 195-200. doi : 10.1016/j.crma.2015.10.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.021/

[1] T. Bonnesen Les Problèmes des Isopérimètres et des Isépiphanes, Gauthier-Villars, Paris, 1929

[2] K.J. Böröczky; E. Lutwak; D. Yang; G. Zhang The log-Brunn–Minkowski inequality, Adv. Math., Volume 231 (2012), pp. 1974-1997

[3] Y.D. Burago; V.A. Zalgaller Geometric Inequalities, Springer-Verlag, Berlin, 1988

[4] M.E. Gage An isoperimetric inequality with applications to curve shortening, Duke Math. J., Volume 50 (1983), pp. 1225-1229

[5] M.E. Gage Curve shortening makes convex curves circular, Invent. Math., Volume 76 (1984), pp. 357-364

[6] M.E. Gage Positive centers and the Bonnesen inequality, Proc. Amer. Math. Soc., Volume 110 (1990), pp. 1041-1048

[7] M.E. Gage; R.S. Hamilton The heat equation shrinking convex plane curves, J. Differ. Geom., Volume 23 (1986), pp. 69-96

[8] M. Green; S. Osher Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves, Asian J. Math., Volume 3 (1999), pp. 659-676

[9] M. Henk; M.A. Hernández Cifre On the location of roots of Steiner polynomials, Bull. Braz. Math. Soc., Volume 42 (2011), pp. 153-170

[10] P.L. Huang; S.L. Pan; Y.L. Yang Positive center sets of convex curves, Discrete Comput. Geom., Volume 54 (2015), pp. 728-740

[11] M. Jetter Bounds on the roots of the Steiner polynomial, Adv. Geom., Volume 11 (2011), pp. 313-317

[12] M.J. Kaiser The ε-positive center figure, Appl. Math. Lett., Volume 9 (1996), pp. 67-70

[13] R. Osserman Bonnesen-style isoperimetric inequalities, Amer. Math. Mon., Volume 86 (1979), pp. 1-29

[14] L.A. Santaló Integral Geometry and Geometric Probability, Cambridge University Press, Cambridge, 2004

[15] R. Schneider Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia of Mathematics and Its Applications, vol. 44, Cambridge University Press, Cambridge, 1993

Cited by Sources:

☆ This work is supported by the National Science Foundation of China (No. 11171254) and a grant of “The First-class Discipline of Universities in Shanghai”.

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