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:
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.