  We study the problems of asymptotic and approximate consensus in which agents
have to get their values arbitrarily close to each others' inside the convex
hull of initial values, either without or with an explicit decision by the
agents. In particular, we are concerned with the case of multidimensional data,
i.e., the agents' values are $d$-dimensional vectors. We introduce two new
algorithms for dynamic networks, subsuming classical failure models like
asynchronous message passing systems with Byzantine agents. The algorithms are
the first to have a contraction rate and time complexity independent of the
dimension $d$. In particular, we improve the time complexity from the
previously fastest approximate consensus algorithm in asynchronous message
passing systems with Byzantine faults by Mendes et al. [Distrib. Comput. 28]
from $\Omega\!\left( d \log\frac{d\Delta}{\varepsilon} \right)$ to $O\!\left(
\log\frac{\Delta}{\varepsilon} \right)$, where $\Delta$ is the initial and
$\varepsilon$ is the terminal diameter of the set of vectors of correct agents.
