Automatic Convexity of Optimal Spiral Strategies

Definitions and mathematical statement

The supplied record asks whether every optimal spiral-like fire-containment strategy necessarily satisfies both local convexity and assumption $(\mathrm{A1})$, described in the clean statement as an initial-angle condition.

To display the intended quantifiers, write $\mathcal S$ schematically for the class of admissible spiral-like strategies intended in the source report. Write $O(s)$ for optimality, $L(s)$ for the required local convexity, and $A_1(s)$ for satisfaction of the source's assumption $(\mathrm{A1})$. The extracted question is whether \[ \forall s\in\mathcal S,\qquad O(s)\Longrightarrow\left(L(s)\land A_1(s)\right). \] These symbols are placeholders for missing source definitions, not independently specified mathematical predicates. The supplied record does not define the fire-propagation model, admissibility constraints, spiral-like strategies, optimization criterion, relevant notion of local convexity, or the actual condition $(\mathrm{A1})$. Consequently, it determines the intended universal implication but does not supply enough information for a self-contained precise mathematical statement. Those definitions and their associated hypotheses must be recovered from the cited report before formal mathematical review.

The quoted report passage expresses a belief that local convexity and $(\mathrm{A1})$ hold automatically for optimal spiral-like strategies and calls this an open question; it does not assert the implication as a theorem.

Short English statement

Must every optimal spiral-like fire-containment strategy automatically be locally convex and satisfy the source's initial-angle assumption $(\mathrm{A1})$? The supplied record omits the model, optimality criterion, and definitions needed to make this question precise.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005731$, title "Automatic Convexity of Optimal Spiral Strategies," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2023.
• Supplied report citation: Control Methods in Hyperbolic PDEs (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/52, pp. 2941--3011. Also listed among the supplied literature sources as "Oberwolfach Report 52/2023." https://doi.org/10.4171/owr/2023/52. The record attributes the extracted question to a passage in this report.
• Bianchini and Zizza, Existence of spiral strategies for blocking fire spreading, described in the supplied assessment as a 2025 preprint. https://arxiv.org/abs/2508.05324.
• Supplied literature assessment, checked on 2026-08-22: the preprint proves a sharp threshold and constructs an optimal spiral-like barrier, but does not explicitly prove the extracted automatic-local-convexity and $(\mathrm{A1})$ implication; the exact claim is recorded as unresolved. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the problem "open" and its statement status corrected_verified, with a supplied assertion that the clean statement was checked against the official report. These are registry claims; neither that source verification nor the mathematical status has been independently verified here.