The logarithmic spiral is optimal for shoreline search: a computer-assisted proof

Abstract

A ship starts at a point of the plane and moves at unit speed; it has to reach an unknown straight line, of which neither the distance nor the direction is known. The competitive ratio of a path is the supremum, over all lines, of the time at which the line is reached divided by its distance. Baeza-Yates, Culberson and Rawlins conjectured that a logarithmic spiral, with ratio $C_{\mathrm{sp}} = 13.8111351794611\ldots$, is optimal. We give a computer-assisted proof. Paths are arbitrary: the distance from the start and the polar angle may both decrease. The proof lifts the set of found directions to the universal cover of the circle, where unfolding the polar angle can only increase it (Kneser-Poulsen on the line); a bookkeeping inequality with a monotone final source then bounds the covered measure by the reward of a three-state relaxed control problem, in which inward motion is an ordinary control and excursions below the guaranteed disk are impulses. An explicit $C^1$ storage function, a tensor cubic B-spline plus a closed-form term, satisfies the dissipation inequalities of that problem at the spiral's level and is tight only at the spiral; this is verified with about $10^6$ boxes of Arb ball arithmetic, an exact jet and an interval Hessian at the spiral.