I've been trying to understand this paper:
"Random Walks and An $O^*(n^5)$ Volume Algorithm for Convex Bodies", Ravi Kannan, Laszlo Lovasz, Miklos Simonovits.
Motivation:
The paper is about estimating the volume of convex bodies using a random walk. We need a particular inequality to show that the random walk distributes well inside the convex body. The proof uses an interesting technique called localization in which we reduce to proving an inequality on a particular line segment. I'm stuck on one of the intermediate steps. The trouble with my question is that you may have to backtrack into the paper to fully understand the terms of the equation.
Definition of terms:
On page 20 we define a measure on $[0,1]$ by $\mu(T) = \int_T g(t)^{n-1} l(t) \, dt$ where $l, g : [0,1] \rightarrow \mathbb{R}^{+}$ and $g$ is a linear function.
We have another positive function $h$ and sets $J_1, J_2 \in [0,1]$ which partition $[0,1]$ such that $\int_0 ^1 g^{n-1} h \, dt \lt \frac{\delta}{5d \sqrt{n}} \min_{i=1,2} \int_{J_i} g^{n-1} l \,dt$
We also define $J_i' = \{t \in J_i \, | \, h(t) \lt \frac{1}{9} l(t)\}$
And $B = [0,1] - (J_1' \cup J_2')$
My question:
I understand that $\mu(B) \lt \frac{9}{5} \frac{\delta}{d \sqrt{n}} \min\{\mu(J_1), \mu(J_2)\}$ but why does inequality (7) follow, i.e. that
$$\mu(B) \lt \frac{2 \delta}{d \sqrt{n}} \min\{\mu(J_1'), \mu(J_2')\}\; (7)$$
I'm getting a little overwhelmed with the amount of different sets and functions in consideration and its not clear which parts are relevant here. It would suffice to show $\mu(J_i) \lt 10/9 \mu(J_i')$ but at this point it's not even clear that $J_i' \ne \emptyset$.