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