On convex body chasing

A player moving in the plane is given a sequence of instructions of the following type: at step i a planar convex set F_i is specified, and the player has to move to a point in F_i. The player is charged for the distance traveled. We provide a strategy for the player which is competitive, i.e., for any sequence F_i the cost to the player is within a constant (multiplicative) factor of the "off-line" cost (i.e., the least possible cost when all F_i are known in advance). We conjecture that similar strategies can be developed for this game in any Euclidean space and perhaps even in all metric spaces. The analogous statement where convex sets are replaced by more general families of sets in a metric space includes many on-line/off-line problems such as the k-server problem; we make some remarks on these more general problems.