2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63370This 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.14 pages latex, 2 figuresGeometric TopologyDifferential GeometryMetric Geometry53A10 (primary); 52B60, 57Q15 (secondary)text