# Understanding Paul Lévy Theorem [Riemannian Geometry]

In Gromov's paper regarding this proof, he says that by taking the best possible $H$ , we can get the following result: Let $k>0 , x$ be such that $1-x \leq K \int_0^{\infty} J_{k,H} (t) dt$ and $x \leq K \int_0^{\infty} J_{k,-H} (t) dt$ for some constant $K$ . He then says that this implies that $K \geq \phi \circ \Phi^{-1}(x)$ , where:

$\phi(t) = \frac{sin^{n-1} ( \sqrt{k} t) }{\int_0^{\pi/\sqrt{k}}sin^{n-1} (\sqrt{k} s ) ds}$

$\Phi(t)=\int_0 ^t \phi(s)ds$ , $J_{k,H} (t) = ( sLk ' (t) + \frac{H}{n-1} s_k(t) ) _+ ^{n-1}$ $s_k(t) = \frac{sin(\sqrt{k} t ) }{\sqrt{k} }$ .

does someone have an idea about this conclusion? How did he deduce this fact?

thanks !

[http://www.ihes.fr/~gromov/PDF/11[33].pdf ]

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@Jeremy, do you think it might be possible to understand your question? (Better say what is the place in the paper your are trying to understand.) –  Anton Petrunin Jul 24 '12 at 12:26
Or at least provide more context for these constants, functions, and formulas. –  Deane Yang Jul 24 '12 at 12:38
I really cannot find these formulas in the paper you quoted. What is the precise page number? –  YangMills Jul 28 '12 at 16:00