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> *The question turns out quite long and maybe a bit vague, I apologize in advance for that.*
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I am currently trying to understand Cheeger and Colding proof of the almost splitting theorem. Currently I am having trouble understanding how they use the so-called segment inequality. Recall that the segment inequality is the following estimate :




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Theorem (Segment inequality, Cheeger and Colding)
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Let $(M^n,g)$ be a Riemannian Manifold with $Ric\geq -(n-1)g$. Let $B\_x$ and $B\_y$ be two open sets in $M$. Let $f$ be a nonnegative function on $M$, for almost every pair $(x,y)$ in $M^2$, there is a unique unit speed minimizing geodesic $\gamma$ from $x$ to $y$. Set $\mathcal{F}\_f(x,y)=\int\_0^Lf\circ\gamma(s)ds$. Set $D=\sup\_{x\in B\_x,y\in B\_y}d(x,y)$. Then:


$$\int\_{B\_x\times B\_y}\mathcal{F}\_f(x,y)dx\,dy\leq CD(|B\_x|+|B\_y|)\int\_M f(w)dw $$
where $|A|$ denotes the riemannian volume of a set $A$, $C$ is an explicit constant depending only the dimension and the integrals are computed with respect to the riemannian volume element.




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Setting
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Then fix $\varepsilon>0$ set $B\_x=B(x\_0,\varepsilon)$ and $B\_y=B(y\_0,\varepsilon)$. The previous estimate shows that one can find $x^\*\in B(x\_0,\varepsilon)$ and $y^\*\in B(y\_0,\varepsilon)$ such that:


$$ \mathcal{F}\_f(x^\*,y^\*)\leq CD\frac{|B(x\_0,\varepsilon)|+|B(x\_0,\varepsilon)|}{|B(x\_0,\varepsilon)||B(y\_0,\varepsilon)|}\int\_M f(w)dw $$


When proving the almost splitting theorem, one has an upper on $\int\_M f$ for some geometrically relevant function $f$ which can be made has small as wanted using some geometric freedom we have in the context, and one needs for given points $x$ and $y$ to find $x^\*\in B(x\_0,\varepsilon)$ and $y^\*\in B(y\_0,\varepsilon)$ such that $ \mathcal{F}\_f(x^\*,y^\*) $ is as small as wanted, in a way depending only on the geometric parameters at hand.


The problem I have is that the upper bound I have doesn't seem to do the job as the lower bound on the Ricci curvature will not give me a universal upper bound on $\frac{|B(x\_0,\varepsilon)|+|B(x\_0,\varepsilon)|}{|B(x\_0,\varepsilon)||B(y\_0,\varepsilon)|}$. I would like to understand why one can bound this.


It would be easy to remedy this by adding an a priori lower bound on the volume of balls (non collapsing assumption), but it is not part of the hypothesis.




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Question
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My question here is really "what's going on here ? did I miss something obvious ?", but to phrase it a bit more formally :



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> How can one bound $\mathcal{F}\_f(x^\*,y^\*)$ without a non collapsing assumption ?
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*References :* 


These interrogations came from reading *Degeneration of Riemannian Metrics under Ricci curvature bounds* by J. Cheeger, specifically page 50 of these notes. I looked at the Cheeger Colding paper on the almost Rigidity of warped products but couldn't find an answer.



Applying Cheeger and Colding segment inequality
I think that the point is that you need to use the Bishop-Gromov inequality (assuming $\mathrm{ricci}\_g\ge -(n-1)\kappa^2g$ :
$$\frac{\mathrm{vol}\_g B(x,R)}{\mathrm{vol}\_g B(x,\epsilon)}\le \frac{v\_n(\kappa, R)}{v\_n(\kappa,\epsilon)}$$
Hence you will get some points $x^\*\in B(x\_0,\epsilon), y^\*\in B(y\_0,\epsilon)$ with
$$\mathcal{F}\_f(x^\*,y^\*)\le C(n,\kappa,R,\epsilon)\frac{1}{\mathrm{vol} B(o,10000R)} \int\_{B(o,10000R)}f $$


You will find some hints in Gallot's Bourbaki seminar (see the remarque 2.8 :<http://archive.numdam.org/ARCHIVE/SB/SB_1997-1998__40_/SB_1997-1998__40__7_0/SB_1997-1998__40__7_0.pdf> )


