Monotonicity in the higher dimension

Determine whether, for each fixed dimension n≥1, the Gromov–Hausdorff distance d_GH(B^m,B^n) between Euclidean unit balls is nondecreasing as the higher dimension m ranges over integers greater than n.

Background

The paper establishes lower and upper bounds for d_GH(Bm,Bn), including a lower bound depending only on n, a covering-radius lower bound depending on both dimensions, and the strict upper bound d_GH(Bm,Bn)<1. These results do not determine how the distance varies with m for fixed n. The proposed problem asks whether increasing the dimension of the higher-dimensional ball can ever decrease the Gromov–Hausdorff distance.

References

For a fixed dimension $n\geq 1$, is $d_GH(Bm,Bn)$ a nondecreasing function of $m>n$?

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

How do $d_GH(Bm,Bn)$ and $d_GH(S{m-1},S{n-1})$ relate for $m>n$, where the spheres are equipped with the restriction of the Euclidean metric rather than the geodesic metric (see)? For a map $f\colon S{m-1} \rightarrow S{n-1}$, consider the radial extension $\bar{f}\colon Bm \rightarrow Bn$ defined as $\bar{f}(p)= |p| f(\frac{p}{|p|})$ for $p\neq 0$ and $\bar{f}(0)=0$; do connections between $dis(f)$ and $dis(\bar{f})$ yield a relationship between these Gromov--Hausdorff distances?

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

Does $d_H(K_{n+2}(B{n+1}),K_{n+2}(Bn))=d(M_{n+2},K_{n+2}(Bn))$?

— Gromov--Hausdorff Distance Between Euclidean Unit Balls  (2609.09652 - Adams et al., 9 Sep 2026) in Question, Section 8 (Using curvature sets of balls to lower bound d_GH)