Polynomial-in-d upper bound for increasing-chord curves in Euclidean space

Determine whether the maximum possible length of a curve γ in R^d with the increasing chord property (i.e., |bc| ≤ |ad| whenever a, b, c, d appear in that order along γ), normalized by the endpoint distance |st|, admits an upper bound that is polynomial in the dimension d; equivalently, improve the exponential upper bound given in Theorem 1(ii), |γ| ≤ 2·(e/2·(d+4))^{d−1}·|st|, to a bound of the form poly(d)·|st| for d ≥ 3.

Background

The paper proves the first explicit upper bounds in higher dimensions for the length of curves with the increasing chord property and gives a subquadratic algorithmic test in higher dimensions. In particular, Theorem 1(ii) establishes an upper bound of order 2·(e/2·(d+4)){d−1} on the ratio |γ|/|st| in Rd for d ≥ 3.

The authors note that this exponential dependence on d is likely far from optimal and pose whether a polynomial dependence on d is possible, which would significantly tighten the asymptotic behavior of the best upper bound for such curves.

References

Some interesting questions remain: The upper bound in Theorem~\ref{thm:d-space} on the maximum length of a curve with increasing chords in $Rd$ is surely far from the truth. Can one deduce a bound that is polynomial in $d$?

— Arcs with increasing chords in $\mathbf{R}^d$  (2509.01580 - Dumitrescu et al., 1 Sep 2025) in Section 5, Concluding remarks (item 1)