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We present a new class of curves which are self-approaching in the following sense. For any three consecutive points a, b, c on the curve the point b is closer to c than a to c. This is a generalisation of curves with increasing chords which are self-approaching in both directions. We show a tight upper bound of 5.3331. .. for the length of a self-approaching curve over the distance between its endpoints. Keywords. Curves with increasing chords, self-approaching curves, convex hull, detour, arc length.

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Searching for a point in the plane is a well-known search game problem introduced in the early eighties. The best known search strategy is given by a spiral and achieves a competitive ratio of 17.289 . . . It was shown by Gal [14] that this strategy is the best strategy among all monotone and periodic strategies. Since then it was unknown whether the given strategy is optimal in general. This paper settles this old open fundamental search problem and shows that spiral search is indeed optimal. The given problem can be considered as the continuous version of the well-known m-ray search problem and also appears in several non-geometric applications and modifications. Therefore the optimality of spiral search is an important question considered by many researchers in the last decades. We answer the logarithmic spiral conjecture for the given problem. The lower bound construction might be helpful for similar settings, it also simplifies existing proofs on classical m-ray search.