Competitively Chasing Convex Bodies

Let be a family of sets in some metric space. In the -chasing problem, an online algorithm observes request sequence and responds (online) by giving points these sets. The movement cost is distance between consecutive such points. competitive ratio worst case (over sequences) total smallest one could have achieved knowing advance sequence. said to chaseable if there exists with finite ratio. 1991, Linial Friedman conjectured that convex Euclidean space chaseable. We prove this conjecture.