Geometry of Cuts and Metrics
M.M. Deza and M. Laurent,
Springer, 1997
To correct Proposition 10.2.1, one must replace (iii) by:
(iii') For every positive semidefinite matrix $B$ with $Be=0$, the inequality (10.2.5) holds.
The implication (iii') ==> (ii) is now clear. For the implication (ii) ==> (iii'), use the fact that, if $D$ is a matrix of negative type and $B$ is a positive semidefinite matrix with $Be=0$, then $\sum_{i,j}B_{ij}D_{ij} \le 0$. To see it, write $B=bb^T + cc^T + ... $ for some vectors $b,c,...$ with $b^Te=c^Te=...=0$, and note that $\sum_{i,j}b_ib_jD_{ij} \le 0$ for any real vector $b$ with $b^Te=0$ (using a continuity argument).