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DEFORMING A CONVEX DOMAIN INTO A DISK BY KLAIN’S CYCLIC REARRANGEMENT

Part of: General convexity

Published online by Cambridge University Press:  20 February 2018

YUNLONG YANG Open the ORCID record for YUNLONG YANG [Opens in a new window]  and
DEYAN ZHANG
YUNLONG YANG*
Affiliation:
Department of Mathematics, Dalian Maritime University, Dalian, 116026, PR China email ylyang@dlmu.edu.cn
DEYAN ZHANG
Affiliation:
School of Mathematical Sciences, Huaibei Normal University, Huaibei, 235000, PR China email zhangdy8005@126.com
*
✉ylyang@dlmu.edu.cn
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Abstract

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For a convex domain, we use Klain’s cyclic rearrangement to obtain a sequence of convex domains with increasing area and the same perimeter which converges to a disk. As a byproduct, we give a proof of the classical isoperimetric inequality in the plane.

Keywords

convex domaincyclic rearrangementisoperimetric inequalityo-symmetric

MSC classification

Primary: 52A10: Convex sets in $2$ dimensions (including convex curves)
Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 2 , April 2018 , pp. 313 - 319
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author is supported in part by the Fundamental Research Funds for the Central Universities (no. 3132017046) and the Doctoral Scientific Research Foundation of Liaoning Province (no. 20170520382). The second author is supported in part by the National Science Foundation of China (no. 11671298) and the Key Project of Natural Science Research in University in Anhui Province (no. KJ2016A635).

References

Andrews, B., ‘Gauss curvature flow: the fate of the rolling stones’, Invent. Math. 138 (1999), 151–161.CrossRefGoogle Scholar
Barchiesi, M., Cagnetti, F. and Fusco, N., ‘Stability of the Steiner symmetrization of convex sets’, J. Eur. Math. Soc. 15 (2013), 1245–1278.Google Scholar
Bonnesen, T. and Fenchel, W., Theorie der Konvexen Körper (Springer, Berlin, 1974).Google Scholar
Ferone, V. and Mercaldo, A., ‘Neumann problems and Steiner symmetrization’, Comm. Partial Differential Equations 30 (2005), 1537–1553.Google Scholar
Gage, M. E., ‘An isoperimetric inequality with applications to curve shortening’, Duke Math. J. 50 (1983), 1225–1229.Google Scholar
Gage, M. E., ‘Curve shortening makes convex curves circular’, Invent. Math. 76 (1984), 357–364.Google Scholar
Gage, M. E. and Hamilton, R. S., ‘The heat equation shrinking convex plane curves’, J. Differ. Geom. 23 (1986), 69–96.Google Scholar
Grayson, M., ‘The heat equation shrinks embedded plane curves to round points’, J. Differ. Geom. 26 (1987), 285–314.Google Scholar
Huisken, G., ‘Flow by mean curvature of convex surfaces into spheres’, J. Differ. Geom. 20 (1984), 237–266.Google Scholar
Klain, D. A., ‘An error estimate for the isoperimetric deficit’, Illinois J. Math. 49 (2005), 981–992.Google Scholar
Klain, D. A., ‘Steiner symmetrization using a finite set of directions’, Adv. Appl. Math. 48 (2012), 340–353.Google Scholar
Lin, Y. J. and Leng, G. S., ‘The Steiner symmetrization of log-concave functions and its applications’, J. Math. Inequal. 7 (2013), 669–677.Google Scholar
Lyusternik, L. A., Convex Figures and Polyhedra (D. C. Heath and Company, Boston, 1966).Google Scholar
Yaglom, I. M. and Boltyanski, V. G., Convex Figures (Holt, Rinehart and Winston, New York, 1960), translated by P. J. Kelly and L. F. Walton.Google Scholar
Ye, D. P., ‘On the monotone properties of general affine surface areas under the Steiner symmetrization’, Indiana Univ. Math. J. 63 (2014), 1–19.Google Scholar