Order of magnitude of maximal arclength for increasing-chord curves in Euclidean d-space

Determine the asymptotic order of magnitude, as a function of the dimension d, of the maximum possible arclength of a continuous curve in Euclidean R^d that satisfies the increasing chord property, when the curve’s endpoints are at unit Euclidean distance apart; equivalently, determine the growth rate of C_d = sup{ arclength(f) : f is a curve in R^d with the increasing chord property and ||f(1)−f(0)||_2 = 1 }.

Background

A curve satisfies the increasing chord property if for any four points a, b, c, d in this order along the curve, the distance ||a−d|| is at least ||b−c||. In the Euclidean plane (d = 2), Rote determined the sharp constant 2π/3 for the ratio of arclength to endpoint distance.

For d ≥ 3 in Euclidean space, the situation is unclear: beyond an upper bound in R3 due to Rote, the dependence on dimension remains unresolved. This paper extends results to strictly convex normed planes and presents examples in other norms (e.g., the ℓ∞ norm) where the maximum length can grow exponentially with dimension, highlighting the contrast with the unknown Euclidean case.

References

In particular, nothing is known of the order of magnitude of the maximum arclength of these curves in $d$-dimensional Euclidean space, though the above example may indicate that the answer might be linear.

— Curves with increasing chords in normed planes (2509.02312 - Lángi et al., 2 Sep 2025) in Section 5 (Additional remarks and questions), paragraph 1