Affine isoperimetric inequalities for piecewise linear surfaces

Details Affine isoperimetric inequalities for piecewise linear surfaces Affine isoperimetric inequalities for piecewise linear surfaces

2002-02-18

arxiv.org/abs/math/0202179v1

Mathematics

Geometric Topology

14 pages latex, 2 figures

Scientific paper

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.

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