In the Convex Body Chasing problem, we are given an initial point @math in @math and an online sequence of @math convex bodies @math . When we receive @math , we are required to move inside @math . Our goal is to minimize the total distance travelled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an @math lower bound on the competitive ratio, and conjectured that a competitive ratio depending only on d is possible. However, despite much interest in the problem, the conjecture remains wide open. We consider the setting in which the convex bodies are nested: @math . The nested setting is closely related to extending the online LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints. Moreover, this setting retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture. In this work, we give the first @math -competitive algorithm for chasing nested convex bodies in @math .
A player moving in the plane is given a sequence of instructions of the following type: at stepi a planar convex setFi is specified, and the player has to move to a point inFi. The player is charged for the distance traveled. We provide a strategy for the player which is competitive, i.e., for any sequenceFi the cost to the player is within a constant (multiplicative) factor of the "off-line" cost (i.e., the least possible cost when allFi 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.