Speed Convergence in Convex Hull Avoiding Walks

Definitions and mathematical statement

Fix a dimension $d$ in the intended model's admissible range, which is not specified in the supplied record. Let $k$ denote its finite-memory parameter. Write $v_{d,k}$ for the quantity called the speed of the finite-memory convex-hull-avoiding process, and write $v_{d,\infty}$ for the conjectural speed of its infinite-memory counterpart.

Source question. Does the finite-memory speed converge to the conjectural infinite-memory speed, that is, \[ \lim_{k\to\infty} v_{d,k}=v_{d,\infty}, \] with $d$ fixed and $k$ tending to infinity through the admissible memory parameters?

Unresolved specifications. The supplied record does not define the processes, their transition rules, their initial conditions, the precise memory convention, or the admissible dimensions and memory parameters. It also does not define speed or specify the probabilistic limiting statement by which a speed is obtained. Thus the symbols above retain their source-designated meanings; a self-contained model definition cannot be reconstructed from this record alone. In particular, $v_{d,\infty}$ is explicitly conjectural: its existence, determinism, and positivity are not assumed to be established. The record does not clarify whether existence of this speed is part of the question or a prerequisite for the displayed convergence assertion.

Short English statement

For a fixed admissible dimension $d$, do the speeds of the finite-memory convex-hull-avoiding walks converge, as memory grows, to the conjectural speed of the infinite-memory walk? The model definitions, dimension range, and meaning of speed require clarification from the cited sources.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004542$, titled Speed Convergence in Convex Hull Avoiding Walks; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. No individual proposer is identified. The supplied proposed year is 2020.
• Supplied report citation: Stochastic Processes under Constraints (2021), Oberwolfach Reports, pp. 1601--1656, DOI 10.4171/owr/2020/32. https://doi.org/10.4171/owr/2020/32. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference, labeled Finite-memory convex-hull walk: https://doi.org/10.1016/j.indag.2019.11.002. No authors or further bibliographic details are supplied.
• Supplied literature reference, labeled OWR report: https://doi.org/10.4171/owr/2020/32.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports finite-memory ballisticity as proved, describes infinite-memory ballisticity as conjectural, and identifies the speed-convergence assertion as Conjecture 5. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.