Search with segment visibility

\(k \ge 2\) robots of unit speed start together at the origin \(O\) of the plane, share a coordinate frame and a clock, and move deterministically. A stationary target lies at an unknown point \(x \ne O\) at distance \(d = \|x\| > 0\). The target is found at the first time \(t\) at which \(x\) lies on the closed segment joining the positions of two robots at time \(t\); robots learn nothing about the target before. Depending on the setting the robots know \(d\) or not; they never know the direction of \(x\). Time is the competitive ratio: the supremum, over targets, of the time until the target is found divided by \(d\).

Measures: Competitive ratio, worst-case time until the target lies on a segment between two robots, divided by its distance \(d\) from the origin.