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 1/21/\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 1/21/\sqrt{2}. They explicitly identify determining whether the contraction rate can be improved further as a future research question.

References

There are several interesting questions that remain for future work. The most natural question is whether the contraction rate can be improved even further. Note that in one dimension, the contraction rate is $1/2$ and it is tight. In two or more dimensions, there is still a gap between $1/2$ and $1/\sqrt{2}$ that should be investigated further.

— Faster Convergence of Multidimensional Approximate Agreement via Smallest Enclosing Balls  (2609.01490 - Melnyk, 1 Sep 2026) in Section Conclusion

It is not clear if other constructions work to show a general contraction lower bound of \sqrt{\frac{d}{2(d+1)}} for dimension d, but we can construct a similar example in dimension $8$ via octonions with a contraction lower bound of $\sqrt{\frac{8}{2(8+1)}} = \frac 2 3$ for $d \geq 8$.

— New Contraction Bounds for Multidimensional Asymptotic Consensus in Dynamic Networks  (2610.11805 - Aab et al., 8 Oct 2026) in Section 1, Introduction, paragraph preceding “Paper outline”

We are not able to repeat this construction for several rounds, but it shows that the presented algorithm analysis is asymptotically optimal for one round.

— New Contraction Bounds for Multidimensional Asymptotic Consensus in Dynamic Networks  (2610.11805 - Aab et al., 8 Oct 2026) in Section 1, Introduction, paragraph preceding “Paper outline”