```From:           grove@ernie.Berkeley.EDU (Eddie Grove)
Newsgroups:     sci.math
Subject:        Special Triangulations of Convex Polygons
Date:           6 Mar 90 20:01:32 GMT
Reply-To:       grove@ernie.Berkeley.EDU.UUCP (Eddie Grove)
Organization:   University of California, Berkeley
```

```A triangulation of a an n-vertex convex polygon is a partition
by n-3 chords into n-2 triangles.  No 2 of the chords intersect
in the interior of the polygon.

I want to know how many different areas are possible.  Formally,
I want a function t(n) as large as possible (linear would be ideal)
satisfying:

for all n-vertex convex polygons there exists a triangulation
containing at least t(n) triangles of different areas.

I have not seen how to create an example in which 3 triangles
must have the same area.  I would be interested in such an
example, if it exists.

The polygons I am interested in are special.  They lie in planes
in R^3, and all of the vertices are lattice points, within the
m x m x m cube.  The normal directions to the planes can be written
with integer coordinates at most m.  An answer in terms of m would
be helpful, although less satisfying.

Eddie Grove
Eddie Grove
```