Affine Isoperimetric Inequalities for Piecewise Linear Surfaces

We consider affine analogues of the isoperimetric inequality in the sense of piecewise linear (PL) manifolds. Given a closed polygon P having n edges, embedded in R, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface embedded in R having P as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surface having P as its boundary which is immersed in R d and whose interior is disjoint from P . The most interesting case is dimension 3. We use the Seifert surface construction to show that for any polygon embedded in R there exists an embedded orientable triangulated PL surface having at most 7n triangles, whose boundary is a subdivision of P . We complement this with a construction of families of polygons with n vertices for which any such embedded surface requires at least 1 2 n − O(n) triangles. It follows that the (asymptotic) optimal combinatorial isoperimetric constant in dimension 3 lies between 1/2 and 7. We also exhibit families of polygons in R for which Ω(n) triangles are required in any immersed PL surface of the above kind. In contrast, in dimension 2 and in dimensions d ≥ 5 there always exists an embedded locally flat PL disk having P as boundary that contains at most n triangles. In dimension 4 there always exists an immersed locally flat PL disk of the above kind that contains at most 3n triangles An unresolved case is that of embedded PL surfaces in dimension 4, where we establish only an O(n) upper bound.