Croke, Christopher B. Area and the length of the shortest closed geodesic. (English) Zbl 0642.53045 J. Differ. Geom. 27, No. 1, 1-21 (1988). 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 Cited in 4 ReviewsCited in 42 Documents MSC: 53C20 Global Riemannian geometry, including pinching 53C22 Geodesics in global differential geometry 53C40 Global submanifolds Keywords:area; length; isosystolic inequalities; diameter; convex hypersurface Citations:Zbl 0515.53037; Zbl 0563.53035 PDFBibTeX XMLCite \textit{C. B. Croke}, J. Differ. Geom. 27, No. 1, 1--21 (1988; Zbl 0642.53045) Full Text: DOI