Improve the multidimensional contraction rate

Improve the contraction rate of multidimensional Byzantine approximate agreement protocols beyond the achieved rate of $1/\sqrt{2}$, narrowing the gap to the tight one-dimensional rate of $1/2$ for dimensions two and higher.

Background

The paper presents synchronous and asynchronous multidimensional Byzantine approximate agreement algorithms based on the midpoint of the smallest enclosing ball of locally computed safe areas. Both algorithms achieve a contraction rate of $1/\sqrt{2}$, and the paper constructs two-dimensional examples showing that this rate is asymptotically tight for the proposed algorithms.

The authors note that the optimal one-dimensional contraction rate is $1/2$, while in dimensions two and higher a gap remains between $1/2$ and $1/\sqrt{2}$. They explicitly identify determining whether the contraction rate can be improved further as a future research question.