If K is a compact polyhedron in
Euclidean d-space, defined by linear inequalities, βi≥ 0, and if f is a polynomial in d
variables that is strictly positive on K, then f can be expressed as a positive linear
combination of products of members of {βi}. In proving this and subsidiary
results, we construct an ordered ring that is a complete AGL(d,R)-invariant
for K, and discuss some of its properties. For example, the ordered ring
associated to K admits the Riesz interpolation property if and only if it is
AGL(d,R)-equivalent to a product of simplices. This is exploited to show
that certain polynomials are not in the positive cone generated by the set
{βi}.
