The logarithmic spiral is optimal for shoreline search (Baeza-Yates–Culberson–Rawlins conjecture)

Every search path has competitive ratio at least $C_{\mathrm{sp}} = 13.811135179\ldots$, so the logarithmic spiral is optimal and the optimal ratio is exactly $C_{\mathrm{sp}}$. Paths are arbitrary: the distance from the start and the polar angle may both decrease. The proof is computer-assisted: a reduction to a three-state relaxed control problem, closed by an explicit $C^1$ storage function whose dissipation inequalities are verified in interval arithmetic and are tight only at the spiral.

Every search path has competitive ratio at least $C_{\mathrm{sp}} = 13.811135179\ldots$, so the logarithmic spiral is optimal and the optimal ratio is exactly $C_{\mathrm{sp}}$. Paths are arbitrary: the distance from the start and the polar angle may both decrease. The proof is computer-assisted: a reduction to a three-state relaxed control problem, closed by an explicit $C^1$ storage function whose dissipation inequalities are verified in interval arithmetic and are tight only at the spiral.