  Motivated by a path planning problem we consider the following procedure.
Assume that we have two points $s$ and $t$ in the plane and take
$\mathcal{K}=\emptyset$. At each step we add to $\mathcal{K}$ a compact convex
set that does not contain $s$ nor $t$. The procedure terminates when the sets
in $\mathcal{K}$ separate $s$ and $t$. We show how to add one set to
$\mathcal{K}$ in $O(1+k\alpha(n))$ amortized time plus the time needed to find
all sets of $\mathcal{K}$ intersecting the newly added set, where $n$ is the
cardinality of $\mathcal{K}$, $k$ is the number of sets in $\mathcal{K}$
intersecting the newly added set, and $\alpha(\cdot)$ is the inverse of the
Ackermann function.
