  In this paper, we consider two multi-dimensional Hagselmann-Krause (HK)
models for opinion dynamics. The two models describe how individuals adjust
their opinions on multiple topics, based on the influence of their peers. The
models differ in the criterion according to which individuals decide whom they
want to be influenced from. In the average-based model, individuals compare
their average opinions on the various topics with those of the other
individuals and interact only with those individuals whose average opinions lie
within a confidence interval. For this model, we provide an alternative proof
for the contractivity of the range of opinions and show that the agents'
opinions reach consensus/clustering if and only if their average opinions do
so. In the uniform affinity model agents compare their opinions on every single
topic and influence each other only if, topic-wise, such opinions do not differ
more than a given tolerance. We identify conditions under which the uniform
affinity model enjoys the order-preservation property topic-wise and we prove
that the global range of opinions (and hence the range of opinions on every
single topic) are nonincreasing.
