2002/02/18 by Joel Hass, Jeffrey C. Lagarias, Hass, Joel +1
Mathematics · #52B60 #53A10 (primary) #57Q15 (secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Mathematics and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.DG #math.GT #math.MG #msc:52B60 #msc:53A10 #msc:57Q15
paper · pdf · doi:10.48550/arxiv.math/0202179
14 pages latex, 2 figures
arxiv created 2002/02/18 · openalex publication_date 2002/02/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in Rd having n edges, we give upper and lower bounds for the minimal number of triangles needed to forma triangulated embedded orientable surface in Rd having P as its geometric boundary. The most interesting case is dimension dimension 3, where we give an upper bound of 7 n2 triangles, and a lower bound for some polygons P that require at least 1/2 n2 triangles. In dimension 2 and dimensions 5 and above one needs only O(n) triangles. The case of dimension 4 is not completely resolved.