  This paper considers affine analogues of the isoperimetric inequality in the
sense of piecewise linear topology. Given a closed polygon P embedded in R^d
having n edges, we give upper and lower bounds for the minimal number of
triangles needed to forma triangulated embedded orientable surface in R^d
having P as its geometric boundary. The most interesting case is dimension
dimension 3, where we give an upper bound of 7 n^2 triangles, and a lower bound
for some polygons P that require at least 1/2 n^2 triangles. In dimension 2 and
dimensions 5 and above one needs only O(n) triangles. The case of dimension 4
is not completely resolved.
