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Curve shortening makes convex curves circular. (English) Zbl 0542.53004
The 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
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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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