Let a ship start at the origin and follow an arbitrary deterministic unit-speed path gamma:[0,∞)→R^2. For every straight line L⊂R^2 with d(L):=operatornamedist(0,L)>0, let tau_gamma(L):=inft≥ 0:gamma(t)∈ L, and define the competitive ratio R(gamma):=sup_L tau_gamma(L)/d(L). The conjecture is that inf_gamma R(gamma) is attained by the appropriate logarithmic-spiral search path, and that inf_gamma R(gamma)=C_sp=13.8111351794611….

Baeza-Yates–Culberson–Rawlins shoreline-search conjecture

Let a ship start at the origin and follow an arbitrary deterministic unit-speed path $\gamma:[0,\infty)\to\mathbb{R}^2$ . For every straight line $L\subset\mathbb{R}^2$ with $d(L):=\operatorname{dist}(0,L)>0$ , let $\tau_\gamma(L):=\inf\{t\ge 0:\gamma(t)\in L\}$ , and define the competitive ratio $R(\gamma):=\sup_L \tau_\gamma(L)/d(L)$ . The conjecture is that $\inf_\gamma R(\gamma)$ is attained by the appropriate logarithmic-spiral search path, and that $\inf_\gamma R(\gamma)=C_{\mathrm{sp}}=13.8111351794611\ldots$ .

References

Primary source

arXiv

Additional references

The logarithmic spiral is optimal for shoreline search: a computer-assisted proof — arXiv — Alexander Temerev

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A computer-assisted manuscript claims to prove that the logarithmic spiral is the best possible shoreline-search strategy, but independent verification is still absent.

The conjecture asks whether the logarithmic spiral is optimal among all deterministic search paths, not only spiral-like paths. The underlying conjecture dates to work from 2005 by Baeza-Yates, Culberson, and Rawlins.

Known results

The best known logarithmic spiral has ratio $13.81113517946\ldots$ .
The unconditional lower bound is $12.5937096701246675\ldots$ , without cyclicity, self-similarity, spiral, or angular-monotonicity assumptions.
The 2005 analyses supplied compelling but incomplete evidence for spiral optimality.

September 2026 claimed proof

Alexander Temerev’s arXiv article claims an explicit storage function plus interval-arithmetic verification over roughly one million boxes, matching the spiral lower bound. However, another September 2026 arXiv account records the unequal bounds above and says optimality remains open; the claimed resolution is therefore unverified.

Current status (as of September 2026): A claimed computer-assisted proof would settle the conjecture, but the only independently described bounds remain $12.5937096701246675\ldots\le C_2^*\le13.81113517946\ldots$ , so equality and external validation remain open.

Sources

No solutions have been posted yet.