Gage, M. E. Curve shortening makes convex curves circular. (English) Zbl 0542.53004 Invent. Math. 76, 357-364 (1984). 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 Cited in 2 ReviewsCited in 95 Documents MSC: 53A05 Surfaces in Euclidean and related spaces 52A40 Inequalities and extremum problems involving convexity in convex geometry Keywords:convex curves; deformation of curves; isoperimetric ratio Citations:Zbl 0534.52008 PDFBibTeX XMLCite \textit{M. E. Gage}, Invent. Math. 76, 357--364 (1984; Zbl 0542.53004) Full Text: DOI EuDML 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.