Dimension-dependent improvement of the lower bound

Establish an improved lower bound for d_GH(B^m,B^n) that depends on both the higher dimension m and the lower dimension n, thereby capturing the expected increase in geometric dissimilarity as the dimensional difference grows.

Background

Theorem 2.2 provides a lower bound for d_GH(Bm,Bn) that depends only on n, whereas Proposition 4.1 gives a covering-radius bound depending on m and n. The authors note that the dimension-only bound does not reflect the intuition that balls should become more dissimilar as the gap between their dimensions increases. The unresolved problem is to strengthen the theorem with a genuinely two-parameter lower bound.

References

Can one establish an improved version of Theorem~\ref{thm:general} that is a function of both $m$ and $n$?

— Gromov--Hausdorff Distance Between Euclidean Unit Balls  (2609.09652 - Adams et al., 9 Sep 2026) in Question, Section 6 (Conclusion and open questions), following Theorem 2.2 and Proposition 4.1

Can the methods developed in this paper be extended to bound $d_GH(Bm_p, Bn_p)$ for $\ellp$ unit balls Bn_p = \left{x \in n : \left(\sum_{i=1}n |x_i|p\right){1/p} \leq 1\right} for $1\le p \leq \infty$? Or to ellipsoids ${x \in n : xT A x \leq 1}$ for positive definite matrices $A \neq I$? Since these sets are centrally symmetric ($x \in B$ implies $-x \in B$), what do Borsuk--Ulam approaches yield in this setting?

— Gromov--Hausdorff Distance Between Euclidean Unit Balls  (2609.09652 - Adams et al., 9 Sep 2026) in Question, Section 6 (Conclusion and open questions)

For $n=2$, we conjecture that the configuration that minimizes the distance to $M_4$ places points at $(\pm c,0)$ and $(0,\pm c)$ in $B2$ for $c=\frac{2\sqrt\frac{8}{3}{2+\sqrt{2} \approx0.9566$, which (if true) would yield that $d_GH(B3,B2)$ is at least as large as $\tfrac{1}{2}d(M_4,K_4(B2)){= }\tfrac{1}{2}\lvert\sqrt{\tfrac{8}{3}-2c\rvert \approx0.1401$.

— Gromov--Hausdorff Distance Between Euclidean Unit Balls  (2609.09652 - Adams et al., 9 Sep 2026) in Remark following the curvature-set estimates in Appendix Section 3, before the final curvature-set questions