  Let $P$ be a set of $n$ points in the plane. We consider the problem of
partitioning $P$ into two subsets $P_1$ and $P_2$ such that the sum of the
perimeters of $\text{CH}(P_1)$ and $\text{CH}(P_2)$ is minimized, where
$\text{CH}(P_i)$ denotes the convex hull of $P_i$. The problem was first
studied by Mitchell and Wynters in 1991 who gave an $O(n^2)$ time algorithm.
Despite considerable progress on related problems, no subquadratic time
algorithm for this problem was found so far. We present an exact algorithm
solving the problem in $O(n \log^2 n)$ time and a
$(1+\varepsilon)$-approximation algorithm running in $O(n +
1/\varepsilon^2\cdot\log^2(1/\varepsilon))$ time.
