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Webs of stars or how to triangulate free sums of point configurations. (English) Zbl 1416.52016
If \(P\subseteq\mathbb{R}^d\) and \(Q\subseteq\mathbb{R}^e\) are finite point sets containing the origin as a point that lies in the interior of the convex hull, the free sum \(P\oplus Q\) of \(P\) and \(Q\) is the finite point set in \(\mathbb{R}^{d+e}\) defined as the union of \(P\times\{0\}\) and \(\{0\}\times Q\). The paper classifies the triangulations of free sums \(P\oplus Q\) in terms of the triangulations of the summands \(P\) and \(Q\). As a case study, the methods are applied to triangulations of point sets associated with the vertex-sets of smooth Fano polytopes.
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
52B45 Dissections and valuations (Hilbert’s third problem, etc.)
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
Full Text: DOI
[1] Assarf, B.; Nill, B., A bound for the splitting of smooth Fano polytopes with many vertices, J. Algebraic Combin., 43, 1, 153-172, (2016) · Zbl 1333.52015
[2] Assarf, B.; Joswig, M.; Paffenholz, A., Smooth Fano polytopes with many vertices, Discrete Comput. Geom., 52, 2, 153-194, (2014) · Zbl 1326.14123
[3] Batyrev, V. V., On the classification of toric Fano 4-folds, J. Math. Sci. (New York), 94, 1, 1021-1050, (1999) · Zbl 0929.14024
[4] Cox, D. A.; Little, J. B.; Schenck, H. K., Toric varieties, Graduate Studies in Mathematics, vol. 124, (2011), American Mathematical Society Providence, RI · Zbl 1223.14001
[5] De Loera, J.; Rambau, J.; Santos, F., Triangulations: structures for algorithms and applications, Algorithms and Computation in Mathematics, (2010), Springer-Verlag · Zbl 1207.52002
[6] Gawrilow, E.; Joswig, M., : a framework for analyzing convex polytopes, (Polytopes - Combinatorics and Computation, Oberwolfach, 1997, DMV Sem., vol. 29, (2000), Birkhäuser Basel), 43-73 · Zbl 0960.68182
[7] Herrmann, S.; Joswig, M., Totally splittable polytopes, Discrete Comput. Geom., 44, 1, 149-166, (2010) · Zbl 1198.52011
[8] Hudson, J. F.P., Piecewise linear topology, Mathematics Lecture Note Series, (1969), W.A. Benjamin · Zbl 0189.54507
[9] Kreuzer, M.; Nill, B., Classification of toric Fano 5-folds, Adv. Geom., 9, 1, 85-97, (2009) · Zbl 1193.14067
[10] McMullen, P., Constructions for projectively unique polytopes, Discrete Mathematics, 14, 4, 347-358, (1976) · Zbl 0319.52010
[11] Øbro, M., Classification of smooth Fano polytopes, (2007), University of Aarhus, Available at:
[12] Paffenholz, A., : a database for polytopes and related objects, (2017), Preprint
[13] Pfeifle, J.; Rambau, J., Computing triangulations using oriented matroids, (Algebra, Geometry, and Software Systems, (2003), Springer Berlin), 49-75 · Zbl 1027.52015
[14] Rambau, J., , version 0.17.5. available at:, (2015)
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