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