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 ]