Given a convex region in the plane, and a sweep-line as a tool, what is best
way to reduce the region to a single point by a sequence of sweeps? The problem
of sweeping points by orthogonal sweeps was first studied in [2]. Here we
consider the following \emph{slanted} variant of sweeping recently introduced
in [1]: In a single sweep, the sweep-line is placed at a start position
somewhere in the plane, then moved continuously according to a sweep vector
$\vec v$ (not necessarily orthogonal to the sweep-line) to another parallel end
position, and then lifted from the plane.

The problem of finding ``small'' sets that meet every straight-line which
intersects a given convex region was initiated by Mazurkiewicz in 1916. We call
such a set an {\em opaque set} or a {\em barrier} for that region. We consider
the problem of computing the shortest barrier for a given convex polygon with
$n$ vertices. For general barriers, we present a simple $O(n)$ time
approximation algorithm with ratio $\frac{1}{2}+ \frac{2
+\sqrt{2}}{\pi}=1.5867\ldots$.

The Fermat-Weber center of a planar body $Q$ is a point in the plane from
which the average distance to the points in $Q$ is minimal. We first show that
for any convex body $Q$ in the plane, the average distance from the
Fermat-Weber center of $Q$ to the points of $Q$ is larger than ${1/6} \cdot
\Delta(Q)$, where $\Delta(Q)$ is the diameter of $Q$. This proves a conjecture
of Carmi, Har-Peled and Katz. From the other direction, we prove that the same
average distance is at most $\frac{2(4-\sqrt3)}{13} \cdot \Delta(Q) < 0.3490
\cdot \Delta(Q)$.

We present two new approximation algorithms with (improved) constant ratios
for selecting $n$ points in $n$ unit disks such that the minimum pairwise
distance among the points is maximized.

(I) A very simple $O(n \log{n})$-time algorithm with ratio 0.5110 for
disjoint unit disks. In combination with an algorithm of Cabello \cite{Ca07},
it yields a $O(n^2)$-time algorithm with ratio of 0.4487 for dispersion in $n$
not necessarily disjoint unit disks.

We revisit several maximization problems for geometric networks design under
the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM
Symposium on Computational Geometry, 1993). Given a set of $n$ points in the
plane in general position, compute a longest non-crossing configuration
composed of straight line segments that is: (a) a matching (b) a Hamiltonian
path (c) a spanning tree (d) a Hamiltonian cycle: (i) For the longest
non-crossing Hamiltonian path problem, we give an approximation algorithm with
ratio $\frac{2}{\pi+1} \approx 0.4829$.