  According to a result of Arkin~\etal~(2016), given $n$ point pairs in the
plane, there exists a simple polygonal cycle that separates the two points in
each pair to different sides; moreover, a $O(\sqrt{n})$-factor approximation
with respect to the minimum length can be computed in polynomial time.
  Here the following results are obtained: (I)~We extend the problem to
geometric hypergraphs and obtain the following characterization of feasibility.
Given a geometric hypergraph on points in the plane with hyperedges of size at
least $2$, there exists a simple polygonal cycle that separates each hyperedge
if and only if the hypergraph is $2$-colorable. (II)~We extend the
$O(\sqrt{n})$-factor approximation in the length measure as follows:
  Given a geometric graph $G=(V,E)$, a separating cycle (if it exists) can be
computed in $O(m+ n\log{n})$ time, where $|V|=n$, $|E|=m$.
  Moreover, a $O(\sqrt{n})$-approximation of the shortest separating cycle can
be found in polynomial time.
  Given a geometric graph $G=(V,E)$ in $\mathbb{R}^3$, a separating polyhedron
(if it exists) can be found in $O(m+ n\log{n})$ time, where $|V|=n$, $|E|=m$.
  Moreover, a $O(n^{2/3})$-approximation of a separating polyhedron of minimum
perimeter can be found in polynomial time. (III)~Given a set of $n$ point pairs
in convex position in the plane, we show that a $(1+\varepsilon)$-approximation
of a shortest separating cycle can be computed in time
$n^{O(\varepsilon^{-1/2})}$. In this regard, we prove a lemma on convex polygon
approximation that is of independent interest.
