This work considers the multidimensional approximate agreement problem. In this problem, $n$ parties in a distributed system, up to $t$ of which may be corrupted by a Byzantine adversary, need to output vectors that are close to each other and that lie inside the convex hull of all non-corrupted input vectors. We assume that nodes communicate in a fully-connected authenticated network and analyze synchronous and asynchronous communication models. The focus of this work is on the contraction factor of approximate agreement protocols. The first multidimensional approximate agreement protocols had a contraction rate of $1-1/n$(VG, PODC'13) and $\sqrt[d]{1/2}$(MH, STOC'13). While a rate below $1$ is sufficient for convergence, it is not sufficient for practical applications. To date, the best known convergence rate of approximate agreement algorithms is $\sqrt{7/8}\approx0.935$ (FN, DISC'18), which is achieved through the MidExtremes protocol. This stands in contrast to the lower bound on the convergence rate in the $1$-dimensional setting, which is $1/2$. In this work, we propose BallMidpoint - a novel approximate agreement protocol with a contraction rate of $1/\sqrt{2}\approx 0.707$ in the synchronous and the asynchronous communication models. This algorithm satisfies the optimal resilience under convex validity. The presented contraction rate is achieved by choosing the midpoint of the smallest enclosing ball of the so-called local safe areas, and it is tight for the presented algorithms. Similar to (FN, DISC'18), our algorithm is coordinate-free, and the point inside the safe area can be computed efficiently. This work presents the first multidimensional approximate agreement protocol where the convergence rate is closer to the best known lower bound rather than the upper bound of $1$.

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