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An isoperimetric inequality with applications to curve shortening. (English) Zbl 0534.52008

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

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
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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
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