Hide and seek in a subset of the real line

This paper is concerned with a zero-sum two-person game in which one player, the hider, hides a particle in a given subset, known to both players, of the real line. The seeker, starting at the origin and travelling with speed no larger than one, tries to arrive at the particle as soon as possible. The payoff is the time the seeker requires to arrive at the particle divided by the distance of the particle from the origin. Analytic characterizations are obtained for the value and optimal strategies of the two players. Explicit solutions are obtained for some examples.