Applications of Mathematics, Vol. 65, No. 6, pp. 703-726, 2020


Isocanted alcoved polytopes

María Jesús de la Puente, Pedro Luis Clavería

Received December 27, 2019.   Published online October 21, 2020.

Abstract:  Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their $f$-vectors and checking the validity of the following five conjectures: Bárány, unimodality, $3^d$, flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension $d$, an isocanted alcoved polytope has $2^{d+1}-2$ vertices, its face lattice is the lattice of proper subsets of $[d+1]$ and its diameter is $d+1$. They are realizations of $d$-elementary cubical polytopes. The $f$-vector of a $d$-dimensional isocanted alcoved polytope attains its maximum at the integer $\lfloor d/3\rfloor$.
Keywords:  cubical polytope; isocanted; alcoved; centrally symmetric; almost simple; zonotope; $f$-vector; cubical $g$-vector; unimodal; flag; face lattice; log-concave sequence; tropical normal idempotent matrix; symmetric matrix
Classification MSC:  52B12, 15A80
DOI:  10.21136/AM.2020.0373-19


References:
[1] R. M. Adin: A new cubical $h$-vector. Discrete Math. 157 (1996), 3-14. DOI 10.1016/S0012-365X(96)83003-2 | MR 1417283 | Zbl 0861.52007
[2] R. M. Adin, D. Kalmanovich, E. Nevo: On the cone of $f$-vectors of cubical polytopes. Proc. Am. Math. Soc. 147 (2019), 1851-1866. DOI 10.1090/proc/14380 | MR 3937665 | Zbl 07046511
[3] A. Barvinok: Integer Points in Polyhedra. Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich (2008). DOI 10.4171/052 | MR 2455889 | Zbl 1154.52009
[4] T. Bisztriczky, P. McMullen, R. Schneider, A. I. Weiss (eds.): Polytopes: Abstract, Convex and Computational. NATO ASI Series. Series C. Mathematical and Physical Sciences 440. Kluwer Academic Publishers, Dordrecht (1994). DOI 10.1007/978-94-011-0924-6 | MR 1322054 | Zbl 0797.00016
[5] G. Blind, R. Blind: The cubical $d$-polytopes with fewer than $2^{d+1}$ vertices. Discrete Comput. Geom. 13 (1995), 321-345. DOI 10.1007/BF02574048 | MR 1318781 | Zbl 0824.52013
[6] G. Blind, R. Blind: The almost simple cubical polytopes. Discrete Math. 184 (1998), 25-48. DOI 10.1016/s0012-365x(97)00159-3 | MR 1609343 | Zbl 0956.52008
[7] F. Brenti: Log-concave and unimodal sequences in algebra, combinatorics, and geometry: An update. Jerusalem combinatorics '93. Contemporary Mathematics 178. American Mathematical Society, Providence, 1994, 71-89. DOI 10.1090/conm/178 | MR 1310575 | Zbl 0813.05007
[8] E. Brugallé: Un peu de géométrie tropicale. Quadrature 74 (2009), 10-22. (In French.) DOI 10.1051/quadrature/2009015 | Zbl 1202.14055
[9] E. Brugallé: Some aspects of tropical geometry. Eur. Math. Soc. Newsl. 83 (2012), 23-28. MR 2934649 | Zbl 1285.14069
[10] P. Butkovič: Max-Plus Linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London (2010). DOI 10.1007/978-1-84996-299-5 | MR 2681232 | Zbl 1202.15032
[11] M. J. de la Puente: On tropical Kleene star matrices and alcoved polytopes. Kybernetika 49 (2013), 897-910. MR 3182647 | Zbl 1297.15029
[12] M. J. de la Puente: Distances on the tropical line determined by two points. Kybernetika 50 (2014), 408-335. DOI 10.14736/kyb-2014-3-0408 | MR 3245538 | Zbl 1321.14050
[13] M. J. de la Puente: Quasi-Euclidean classification of alcoved convex polyhedra. Linear Multilinear Algebra 68 (2020), 2110-2142. DOI 10.1080/03081087.2019.1572065 | MR 4160431
[14] M. Develin, F. Santos, B. Sturmfels: On the rank of a tropical matrix. Combinatorial and computational geometry. Mathematical Sciences Research Institute Publications 52. Cambridge University Press, Cambridge, 2005, 213-242. MR 2178322 | Zbl 1095.15001
[15] M. Develin, B. Sturmfels: Tropical convexity. Doc. Math. 9 (2004), 1-27 corrigendum ibid. 9 (2004), 205-206. MR 2054977 | Zbl 1054.52004
[16] B. Grünbaum: Convex Polytopes. John Wiley & Sons, London (1967). DOI 10.1007/978-1-4613-0019-9 | MR 0226496 | Zbl 0163.16603
[17] P. Guillon, Z. Izhakian, J. Mairesse, G. Merlet: The ultimate rank of tropical matrices. J. Algebra 437 (2015), 222-248. DOI 10.1016/j.jalgebra.2015.02.026 | MR 3351964 | Zbl 1316.15030
[18] M. Henk, J. Richter-Gebert, G. M. Ziegler: Basic properties of convex polytopes. Handbook of Discrete and Computational Geometry (J. E. Goodman et al., eds.). CRC Press Series on Discrete Mathematics and Its Applications. CRC Press, Boca Raton, 1997, 243-270. MR 1730169 | Zbl 0911.52007
[19] A. Jiménez, M. J. de la Puente: Six combinatorial clases of maximal convex tropical polyhedra. (2012), 40 pages Available at https://arxiv.org/abs/1205.4162.
[20] W. Jockusch: The lower and upper bound problems for cubical polytopes. Discrete Comput. Geom. 9 (1993), 159-163. DOI 10.1007/BF02189315 | MR 1194033 | Zbl 0771.52005
[21] G. Kalai: The number of faces of centrally-symmetric polytopes. Graphs Comb. 5 (1989), 389-391. DOI 10.1007/BF01788696 | MR 1554357 | Zbl 1168.52303
[22] G. Kalai, M. G. Ziegler (eds.): Polytopes: Combinatorics and Computation. DMV Seminar 29. Birkhäuser, Basel (2000). DOI 10.1007/978-3-0348-8438-9 | MR 1785290 | Zbl 0944.00089
[23] T. Lam, A. Postnikov: Alcoved polytopes I. Discrete Comput. Geom. 38 (2007), 453-478. DOI 10.1007/s00454-006-1294-3 | MR 2352704 | Zbl 1134.52019
[24] G. L. Litvinov, V. P. Maslov (eds.): Idempotent Mathematics and Mathematical Physics. Contemporary Mathematics 377. American Mathematical Society, Providence (2005). DOI 10.1090/conm/377 | MR 2145152 | Zbl 1069.00011
[25] G. L. Litvinov, S. N. Sergeev (eds.): Tropical and Idempotent Mathematics. Contemporary Mathematics 495. American Mathematical Society, Providence (2009). DOI 10.1090/conm/495 | MR 2581510 | Zbl 1172.00019
[26] G. Mikhalkin: What is ... a tropical curve? Notices Am. Math. Soc. 54 (2007), 511-513. MR 2305295 | Zbl 1142.14300
[27] J. Richter-Gebert, B. Sturmfels, T. Theobald: First steps in tropical geometry. Idempotent Mathematics and Mathematical Physics. Contemporary Mathematics 377. American Mathematical Society, Providence, 2005, 289-317. DOI 10.1090/conm/377 | MR 2149011 | Zbl 1093.14080
[28] R. Sanyal, A. Werner, G. M. Ziegler: On Kalai's conjectures concerning centrally symmetric polytopes. Discrete Comput. Geom. 41 (2009), 183-198. DOI 10.1007/s00454-008-9104-8 | MR 2471868 | Zbl 1168.52013
[29] M. W. Schmitt, G. M. Ziegler: Ten problems in geometry. Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination. Springer, New York, 2013, 279-289. DOI 10.1007/978-0-387-92714-5_22 | MR 3087288 | Zbl 1267.52002
[30] M. Senechal (ed.): Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination. Springer, New York (2013). DOI 10.1007/978-0-387-92714-5 | MR 3087288 | Zbl 1267.52002
[31] S. Sergeev: Multiorder, Kleene stars and cyclic projectors in the geometry of max cones. Tropical and Idempotent Mathematics. Contemporary Mathematics 495. American Mathematical Society, Providence, 2009, 317-342. DOI 10.1090/conm/495 | MR 2581526 | Zbl 1179.15033
[32] N. J. A. Sloane: The On-Line Encyclopedia of Integer Sequences. Available at http://oeis.org/ (2020).
[33] D. Speyer: Tropical linear spaces. SIAM J. Discrete Math. 22 (2008), 1527-1558. DOI 10.1137/080716219 | MR 2448909 | Zbl 1191.14076
[34] D. Speyer, B. Sturmfels: The tropical Grassmannian. Adv. Geom. 4 (2004), 389-411. DOI 10.1515/advg.2004.023 | MR 2071813 | Zbl 1065.14071
[35] R. P. Stanley: Log-concave and unimodal sequences in algebra, combinatorics, and geometry. Graph Theory and Its Applications: East and West. Annals of the New York Academy of Sciences 576. New York Academy of Sciences, New York, 1989, 500-535. DOI 10.1111/j.1749-6632.1989.tb16434.x | MR 1110850 | Zbl 0792.05008
[36] N. M. Tran: Enumerating polytropes. J. Comb. Theory, Ser. A 151 (2017), 1-22. DOI 10.1016/j.jcta.2017.03.011 | MR 3663485 | Zbl 06744864
[37] A. Werner, J. Yu: Symmetric alcoved polytopes. Electron. J. Comb. 21 (2014), Article ID 1.20, 14 pages. DOI 10.37236/3646 | MR 3177515 | Zbl 1302.52014
[38] B. Yu, X. Zhao, L. Zeng: A congruence on the semiring of normal tropical matrices. Linear Algebra Appl. 555 (2018), 321-335. DOI 10.1016/j.laa.2018.06.027 | MR 3834207 | Zbl 1396.15022
[39] G. M. Ziegler: Lectures on Polytopes. Graduate Texts in Mathematics 152. Springer, Berlin (1995). DOI 10.1007/978-1-4613-8431-1 | MR 1311028 | Zbl 0823.52002
[40] G. M. Ziegler: Convex polytopes: Extremal constructions and $f$-vector shapes. Geometric Combinatorics. IAS/Park City Mathematics Series 13. American Mathematical Society, Providence, 2007. MR 2383133 | Zbl 1134.52018

Affiliations:   María Jesús de la Puente (corresponding author), Universidad Complutense, Plaza de Ciencias 3, 28040 Madrid, Spain, e-mail: mpuente@ucm.es; Pedro Luis Clavería, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain, e-mail: plcv@unizar.es


 
PDF available at: