Celebratio Mathematica — Hass — Complete Bibliography

Complete Bibliography

Works connected to Jacob E. Goodman

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J. Hass and J. C. Lagarias: “The minimal number of triangles needed to span a polygon embedded in \( \mathbb{R}^d \),” pp. 509–526 in Discrete and computational geometry: The Goodman–Pollack Festschrift. Edited by B. Aronov, S. Basu, J. Pach, and M. Sharir. Algorithms and Combinatorics 25. Springer (Berlin), 2003. MR 2038489 Zbl 1103.52013 incollection

Given a closed polygon \( P \) having \( n \) edges, embedded in \( \mathbb{R}^d \), we give upper and lower bounds for the minimal number of triangles \( t \) needed to form a triangulated PL surface embedded in \( \mathbb{R}^d \) having \( P \) as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surface having \( P \) as its boundary which is immersed in \( \mathbb{R}^d \) and whose interior is disjoint from \( P \). The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert surface construction to show that for any polygon embedded in \( \mathbb{R}^3 \) there exists an embedded orientable triangulated PL surface having at most \( 7n^2 \) triangles, whose boundary is a subdivision of \( P \). We complement this with a construction of families of polygons with \( n \) vertices for which any such embedded surface requires at least \( \frac{1}{2}n^2 - O(n) \) triangles. We also exhibit families of polygons in \( \mathbb{R}^3 \) for which \( \Omega(n^2) \) triangles are required in any immersed PL surface of the above kind. In contrast, in dimension 2 and in dimensions \( d\geq 5 \) there always exists an embedded locally flat PL disk having \( P \) as boundary that contains at most \( n \) triangles. In dimension 4 there always exists an immersed locally flat PL disk of the above kind that contains at most \( 3n \) triangles. An unresolved case is that of embedded PL surfaces in dimension 4, where we establish only an \( O(n^2) \) upper bound. These results can be viewed as providing qualitative discrete analogues of the isoperimetric inequality for piecewise linear (PL) manifolds. In dimension 3 they imply that the (asymptotic) discrete isoperimetric constant lies between \( 1/2 \) and 7.

János Pach	Related
Jacob E. Goodman	Related
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