  The monostatic property of convex polyhedra (i.e. the property of having just
one stable or unstable static equilibrium point) has been in the focus of
research ever since Conway and Guy published the proof of the existence of the
first such object, followed by the constructions of Bezdek and Reshetov. These
examples establish $F\leq 14, V\leq 18$ as the respective \emph{upper bounds}
for the minimal number of faces and vertices for a homogeneous mono-stable
polyhedron. By proving that no mono-stable homogeneous tetrahedron existed,
Conway and Guy established for the same problem the lower bounds for the number
of faces and vertices as $F, V \geq 5$ and the same lower bounds were also
established for the mono-unstable case. It is also clear that the $F,V \geq 5$
bounds also apply for convex, homogeneous point sets with unit masses at each
point (also called polyhedral 0-skeletons) and they are also valid for
mono-monostatic polyhedra with exactly on stable and one unstable equilibrium
point (both homogeneous and 0-skeletons). Here we present an algorithm by which
we improve the lower bound to $V\geq 8$ vertices (implying $f \geq 6$ faces) on
mono-unstable and mono-monostable 0-skeletons. Our algorithm appears to be less
well suited to compute the lower bounds for mono-stability. We point out these
difficulties in connection with the work of Dawson and Finbow who explored the
monostatic property of simplices in higher dimensions.

Metric Geometry;Computational Geometry