Curve shortening makes convex curves circular.

*(English)*Zbl 0542.53004The author considers a one parameter family of closed convex \(C^ 2\) curves in the plane such that, with increasing time parameter, the curves are deformed along their inner normals at a rate proportional to the curvature. In a former paper [Duke Math. J. 50, 1225-1229 (1983; Zbl 0534.52008)] he had shown that the isoperimetric ratio \(L^ 2/A\) decreases, here he proves that it approaches 4\(\pi\) as the enclosed area approaches zero. Under suitable normalization, the curves converge to a unit circle.

Reviewer: R.Schneider

##### MSC:

53A05 | Surfaces in Euclidean and related spaces |

52A40 | Inequalities and extremum problems involving convexity in convex geometry |

##### References:

[1] | Gage, M.E.: An isoperimetric inequality with applications to curve shortening. Duke Math. J.50, (No. 4), 1225 (1983) · Zbl 0534.52008 · doi:10.1215/S0012-7094-83-05052-4 |

[2] | Lay, S.R.: Convex sets and their applications. New York: John Wiley and Sons 1982 · Zbl 0492.52001 |

[3] | Osserman, R.: Bonnesen-style isoperimetric inequalities. Amer. Math. Monthly86 (No. 1) 1 (1979) · Zbl 0404.52012 · doi:10.2307/2320297 |

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