  We consider the combinatorial question of how many convex polygons can be
made by using the edges taken from a fixed triangulation of n vertices. For
general triangulations, there can be exponentially many: we show a construction
that has Omega(1.5028^n) convex polygons, and prove an O(1.62^n) upper bound in
the worst case. If the triangulation is fat (every triangle has its angles
lower-bounded by a constant delta>0), then there can be only polynomially many.
  We also consider the problem of counting convex outerplanar polygons (i.e.,
they contain no vertices of the triangulation in their interiors) in the same
triangulations. In this setting, we get the same exponential bounds in general
triangulations, and lower polynomial bounds in fat triangulations. If the
triangulation is furthermore compact (the ratio between the longest and
shortest distance between any two vertices is bounded), the bounds drop further
to Theta (n^2) for general convex outerplanar polygons, and Theta (n) for fat
convex outerplanar polygons.
