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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.
##### MSC:
 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.)
##### Keywords:
point configuration; triangulation; free sum; Fano polytope
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##### References:
 [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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