The question turns out quite long and maybe a bit vague, I apologize in advance for that.
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 :
Theorem (Segment inequality, Cheeger and Colding)
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.
Setting
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.
Question
My question here is really "what's going on here ? did I miss something obvious ?", but to phrase it a bit more formally :
How can one bound $\mathcal{F}_f(x^*,y^*)$ without a non collapsing assumption ?
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.
