Given a metric space \((S,\rho)\), and a sequence of regions \(F_ i\), \(i=1,\dots,n\), say the plane \(R^ 2\) and a sequence of convex sets, one is supposed to design an algorithm producing a path \(x_ i\), \(i=1,\dots,n\), \(x_ i\in F_ i\), minimizing the cost \(\sum^{n- 1}_{i=1}\rho(x_ i,x_{i+1})\). There is an ``off-line'' version of this problem in which all regions \(F_ i\) are given in advance and an ``on-line'' version where \(F_ i\) is given after the choice of \(x_{i- 1}\). An algorithm which solves the ``on-line'' problem is said to be competitive if its cost is a constant factor within the cost of an optimal algorithm for the ``off-line'' problem. In this case, the family from which sets \(F_ i\) are chosen is called chaseable. The authors show that, although the most natural algorithm turn out not to be competitive, the family of convex sets in the plane is chaseable. They conjecture that the same holds in the \(n\)-dimensional case and raise a problem of finding metric spaces with chaseable families of convex sets.