Given a closed polygon P having n edges, embedded in R^d, we give upper and
lower bounds for the minimal number of triangles t needed to form a
triangulated PL surface in R^d having P as its geometric boundary. The most
interesting case is dimension 3, where the polygon may be knotted. We use the
Seifert suface construction to show there always exists an embedded surface
requiring at most 7n^2 triangles. We complement this result by showing there
are polygons in R^3 for which any embedded surface requires at least 1/2n^2 -
O(n) triangles. In dimension 2 only n-2 triangles are needed, and in dimensions
5 or more there exists an embedded surface requiring at most n triangles. In
dimension 4 we obtain a partial answer, with an O(n^2) upper bound for embedded
surfaces, and a construction of an immersed disk requiring at most 3n
triangles. These results can be interpreted as giving qualitiative discrete
analogues of the isoperimetric inequality for piecewise linear manifolds.Comment: 16 pages, 4 figures. This paper is a retitled, revised version of
math.GT/020217
