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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:

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