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let $a\in(0,1)$, let $f:M\times N\times [0,T)\rightarrow \mathbb{R}$ a $C^{\infty}$ function, with $M,N$ compact manifolds. Assume that we have a closed, convex set $F$ of such functions such that the subset $W=\{f\in F | $ there exist $x,y\in N$ with $f(p,x,t)> af(p,y,t)\}$ is bounded for every $p,t$, then why we can find a positive constant $C$, independent of $p,t$, such that $\min_N f(p,q,t)\geq a\max_N f(p,q,t) - C$

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    We're making progress. Is this proof published? Maybe if you edit your question to include a link or reference, someone will be able to work out an answer from the context.2011-05-10

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