The Number of Triangles Needed to Span a Polygon Embedded in R^d

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dc.creatorHass, Joel
dc.creatorLagarias, Jeffrey C.
dc.date2002-11-22
dc.date.accessioned2026-07-07T04:53:13Z
dc.date.available2026-07-07T04:53:13Z
dc.descriptionGiven 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.
dc.description16 pages, 4 figures. This paper is a retitled, revised version of math.GT/0202179
dc.identifierhttps://arxiv.org/abs/math/0211364
dc.identifierhttp://arxiv.org/abs/math/0211364
dc.identifierpp. 509--526 in: Discrete and Computational Geometry: The Goodman-Pollack Festscrift, (B. Aronov, S. Basu, J. Pach and M. Sharir, Eds.), Springer-Verlag, New York 2003.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/65760
dc.subjectGeometric Topology
dc.subjectDifferential Geometry
dc.subjectMetric Geometry
dc.subject53A10 Primary; 52B60, 57Q15 Secondary
dc.titleThe Number of Triangles Needed to Span a Polygon Embedded in R^d
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