On convex body chasing

A player moving in the plane is given a sequence of instructions of the following type: at stepi a planar convex setF _i is specified, and the player has to move to a point inF _i. The player is charged for the distance traveled. We provide a strategy for the player which is competitive, i.e., for any sequenceF _i the cost to the player is within a constant (multiplicative) factor of the off-line cost (i.e., the least possible cost when allF _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 thek-server problem; we make some remarks on these more general problems.Joel Friedman wishes to acknowledge the National Science Foundation for supporting this research in part under a PYI Grant, CCR-8858788, and IBM for an IBM faculty development award. Nathan Linial's work was supported by a grant from the Binational Science Foundation Israel/US and from the Israel Academy of Sciences.