  In this paper, we study the resilient vector consensus problem in networks
with adversarial agents and improve resilience guarantees of existing
algorithms. A common approach to achieving resilient vector consensus is that
every non-adversarial (or normal) agent in the network updates its state by
moving towards a point in the convex hull of its \emph{normal} neighbors'
states. Since an agent cannot distinguish between its normal and adversarial
neighbors, computing such a point, often called as \emph{safe point}, is a
challenging task. To compute a safe point, we propose to use the notion of
\emph{centerpoint}, which is an extension of the median in higher dimensions,
instead of Tverberg partition of points, which is often used for this purpose.
We discuss that the notion of centerpoint provides a complete characterization
of safe points in $\mathbb{R}^d$. In particular, we show that a safe point is
essentially an interior centerpoint if the number of adversaries in the
neighborhood of a normal agent $i$ is less than $\frac{N_i}{d+1} $, where $d$
is the dimension of the state vector and $N_i$ is the total number of agents in
the neighborhood of $i$. Consequently, we obtain necessary and sufficient
conditions on the number of adversarial agents to guarantee resilient vector
consensus. Further, by considering the complexity of computing centerpoints, we
discuss improvements in the resilience guarantees of vector consensus
algorithms and compare with the other existing approaches. Finally, we
numerically evaluate the performance of our approach through experiments.
