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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)
##### Keywords:
closed geodesic; convex hypersurface
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##### References:
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