Lemma 19.11 in the book (page 529) relates the minimum
depth $d$ of threshold trees for
searching the ``hurt'' inequality in a contradictory sustem $Ax\leq b$ of
linear inequalities with the randomized communication complexity $r$
of this problem:
$r=O(d\log^2n)$

Daniel Apon observed that $\log^2n$ can here be replaced by $\log n$, that is,
we actually have
$r=O(d\log n)$

Originally, Nisan suggested that the results of Feige, Peleg, Raghavan and Upfal
[STOC'90] give more efficient communication protocols for $GT_n$ (the
grater-than function). In this
note, Daniel makes this
observation explicit.