Asymmetric rendezvous search on the circle

The rendezvous search problem asks how two blind searchers in a known searc h region, having maximum speed one, can minimize the expected time needed t o meet. Suppose that two players are placed an arc-distance x is an element of [0, 1/2] apart on a circle of circumference 1, and faced in random dire ctions. If x has a continuous density function h which is either decreasing and satisfies h (1/2) greater than or equal to h (0) /2,or increasing, we determine an optimal rendezvous strategy. Furthermore if h is strictly mono tone, this strategy (which depends in a simple manner on h) is uniquely opt imal. This work extends that of J. V. Howard, who showed for the uniform de nsity h (x) = 2 that 'search and wait' is optimal, with expected search tim e 1/2. We also show that the uniform density is the only counterexample on the circle to S. Gal's conjecture (which he proved for the line) on the non optimality of 'search and wait'.