Having had a quick look, does the following work? Put x= (\sum_i x_i)/3 and put
y(t) = \sum_{i \in R^1_t} x_i = \sum_i \eta_i(t)x_i
and try to substitute these into (3.2).
Observe that
|x| + |y(t)|
= | 1/3 \sum_i x_i | + | \sum_i \eta_i x_i |
\leq | 1/3 \sum_i x_i | + | \sum_i x_i / 3 | + | \sum_i (\eta_i - 1/3)x_i |
&\leq | \sum_i x_i | + | \sum_i (\eta_i - 1/3)x_i |
and this should give what we want on the RHS of the formula you're asking about. It sh