Gage, Michael E. An isoperimetric inequality with applications to curve shortening. (English) Zbl 0534.52008 Duke Math. J. 50, 1225-1229 (1983). For closed convex \(C^ 2\) curves in the plane with length L, area A and curvature function \(\kappa\), the inequality \(\pi \frac{L}{A}\leq \int^{L}_{O}\kappa^ 2ds\) is proved. It is used to show the following: When a convex curve is deformed along its (inner) normal at a rate proportional to its curvature, then the isoperimetric ratio \(L^ 2/A\) decreases. Reviewer: R.Schneider Cited in 2 ReviewsCited in 91 Documents MSC: 52A40 Inequalities and extremum problems involving convexity in convex geometry 53A04 Curves in Euclidean and related spaces 51M25 Length, area and volume in real or complex geometry Keywords:convex curve; deformation of curves; length; area; curvature; isoperimetric ratio PDFBibTeX XMLCite \textit{M. E. Gage}, Duke Math. J. 50, 1225--1229 (1983; Zbl 0534.52008) Full Text: DOI References: [1] R. Osserman, Bonnesen-style isoperimetric inequalities , Amer. Math. Monthly 86 (1979), no. 1, 1-29. JSTOR: · 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.