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Area and the length of the shortest closed geodesic. (English) Zbl 0642.53045
The main result of the paper is that the inequality 31$$\sqrt{A}\geq L$$ is satisfied for any Riemannian metric on the two-sphere where A denotes its area and L the length of its shortest nontrivial closed geodesic. This is closely related to so-called isosystolic inequalities which relate area (or volume) to the length of the shortest not null-homotopic closed geodesic, see M. Gromov [J. Differ. Geom. 18, 1-147 (1983; Zbl 0515.53037)]. The author also shows for any Riemannian metric on the two- sphere that 9D$$\geq L$$ where D denotes its diameter. Furthermore, he proves that there is a constant $$c(n)^ n\sqrt{Vol(M)}\geq L$$ for any closed convex hypersurface in $${\mathbb{R}}^{n+1}$$. The last result was proved independently by A. Treibergs [Invent. Math. 80, 481-488 (1985; Zbl 0563.53035)].
Reviewer: G.Thorbergsson

##### MSC:
 53C20 Global Riemannian geometry, including pinching 53C22 Geodesics in global differential geometry 53C40 Global submanifolds
##### Keywords:
area; length; isosystolic inequalities; diameter; convex hypersurface
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