Determine the sharp dimension dependence of the trajectory-length bound

Determine the optimal dependence on the dimension d in the trajectory-length bound for exact Euclidean ball-proximal point trajectories, given that the lower-bound construction requires growth at least proportional to \(\sqrt d\) while the current upper bound has a larger dimension dependence.

Background

The paper proves that bounded self-contracted sequences in $\mathbb{R}^d$ have finite length, with an explicit upper-bound constant $C_d$ whose dependence on dimension is large. For exact ball-proximal point iterations, this yields a constant-radius termination bound that is linear in the initial distance-to-solution ratio for each fixed dimension.

A polyhedral construction shows that the dimension dependence cannot be eliminated: the trajectory length and iteration count can scale quadratically in the distance-to-radius ratio as the dimension grows, and the relevant lower-bound constant must grow at least as $\sqrt d$. The unresolved issue is to close the gap between this lower bound and the dimension dependence of the available upper bound.