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Estimates of volume by the length of shortest closed geodesics on a convex hypersurface. (English) Zbl 0563.53035
The author estimates the length L of a closed geodesic of minimal length on a closed convex hypersurface \(M^ n\) as follows \[ L^ n\leq (2\pi)^{n-1} \Gamma^ 2(n/2+1/2)Vol(M^ n). \] The main theorem is more general and applies to minimal k-spheres in \(M^ n\) if the usual minimax argument gives a solution - which has only been shown to be the case for \(k=2\) and \(n=3\).
Reviewer: G.Thorbergsson

MSC:
53C22 Geodesics in global differential geometry
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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References:
[1] [B] Busemann, H.: Convex surfaces. New York: Interscience, 1958 · Zbl 0196.55101
[2] [C] Croke, C.: Poincaré’s Problem and the length of the shortest closed geodesic on a convex hypersurface. J. Differ. Geom.17, 595-634 (1982) · Zbl 0501.53031
[3] [G1] Gromov, M.: Structures metriques pour les variétiés riemanniennes, p. 47. Paris: CEDIC 1981
[4] [G2] Gromov, M.: Filling Riemannian manifolds. J. Differ. Geom.18, 1-147 (1983) · Zbl 0515.53037
[5] [S] Santalo, L.: Integral geometry and geometric probability. Reading: Addison-Wesley, 1976
[6] [Sm] Smith, F.R.: On the existence of embedded minimal 2-spheres in the 3-sphere endowed with an arbitrary metric. Thesis, Melbourne (1981)
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