Controlled-radius contractions of Gromov–Hausdorff balls

Construct, or determine the impossibility of constructing, for every compact metric space X, a constant C and sufficiently small radius r, a contraction of the Gromov–Hausdorff ball centered at X into the ball of radius Cr that fixes X throughout the homotopy, and determine whether C can be chosen independently of X.

Background

This question strengthens the preceding ball-contractibility problem by requiring a quantitative control on the radius containing the homotopy image. The desired homotopy should contract the ball of radius r to its center while remaining inside the ball of radius Cr.

The paper also asks whether the multiplicative constant can be universal, even if the admissible small-radius threshold depends on the center.

References

For every $X\inM$, do there exist $C\geq1$ and $r_0>0$ such that, for every $0<r<r_0$, there is a continuous map $H\colon (X,r)\times[0,1]\to (X,\ContractionFactorr)$ satisfying $H(Y,0)=Y$, $H(Y,1)=X$, $H(X,t)=X$ for all $Y\in(X,r)$ and $t\in[0,1]$? Can $C$ be chosen independently of $X$, with $r_0$ still allowed to depend on $X$?

— The topology of Gromov--Hausdorff space  (2609.09639 - Ishiki, 9 Sep 2026) in Question 4, Section 6 (Questions)