Mathematics – Geometric Topology
Scientific paper
2002-11-22
pp. 509--526 in: Discrete and Computational Geometry: The Goodman-Pollack Festscrift, (B. Aronov, S. Basu, J. Pach and M. Shar
Mathematics
Geometric Topology
16 pages, 4 figures. This paper is a retitled, revised version of math.GT/0202179
Scientific paper
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.
Hass Joel
Lagarias Jeffrey C.
No associations
LandOfFree
The Number of Triangles Needed to Span a Polygon Embedded in R^d does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Number of Triangles Needed to Span a Polygon Embedded in R^d, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Number of Triangles Needed to Span a Polygon Embedded in R^d will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-453060