Journal articles: 'Polytopes'

1

Schulte, Egon, and Asia Ivić Weiss. "Free Extensions of Chiral Polytopes." Canadian Journal of Mathematics 47, no. 3 (June 1, 1995): 641–54. http://dx.doi.org/10.4153/cjm-1995-033-7.

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AbstractAbstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If 𝒦 is a chiral polytope with regular facets 𝓕 then among all chiral polytopes with facets 𝒦 there is a universal such polytope 𝓟, whose group is a certain amalgamated product of the groups of 𝒦 and 𝓕. Finite extensions are also discussed.

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Ramanath, Rohan, S. Sathiya Keerthi, Yao Pan, Konstantin Salomatin, and Kinjal Basu. "Efficient Vertex-Oriented Polytopic Projection for Web-Scale Applications." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 4 (June 28, 2022): 3821–29. http://dx.doi.org/10.1609/aaai.v36i4.20297.

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We consider applications involving a large set of instances of projecting points to polytopes. We develop an intuition guided by theoretical and empirical analysis to show that when these instances follow certain structures, a large majority of the projections lie on vertices of the polytopes. To do these projections efficiently we derive a vertex-oriented incremental algorithm to project a point onto any arbitrary polytope, as well as give specific algorithms to cater to simplex projection and polytopes where the unit box is cut by planes. Such settings are especially useful in web-scale applications such as optimal matching or allocation problems. Several such problems in internet marketplaces (e-commerce, ride-sharing, food delivery, professional services, advertising, etc.), can be formulated as Linear Programs (LP) with such polytope constraints that require a projection step in the overall optimization process. We show that in some of the very recent works, the polytopic projection is the most expensive step and our efficient projection algorithms help in gaining massive improvements in performance.

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Fujita, Naoki. "Newton–Okounkov polytopes of flag varieties and marked chain-order polytopes." Transactions of the American Mathematical Society, Series B 10, no. 15 (April 4, 2023): 452–81. http://dx.doi.org/10.1090/btran/142.

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Marked chain-order polytopes are convex polytopes constructed from a marked poset. They give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand–Tsetlin poset of type A A , and realize the associated marked chain-order polytopes as Newton–Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as Newton–Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal projective toric variety corresponding to a marked chain-order polytope. We also construct a specific basis of an irreducible highest weight representation. The basis is naturally parametrized by the set of lattice points in a marked chain-order polytope.

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Hibi, Takayuki, and Nan Li. "Unimodular Equivalence of Order and Chain Polytopes." MATHEMATICA SCANDINAVICA 118, no. 1 (March 7, 2016): 5. http://dx.doi.org/10.7146/math.scand.a-23291.

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Order polytope and chain polytope are two polytopes that arise naturally from a finite partially ordered set. These polytopes have been deeply studied from viewpoints of both combinatorics and commutative algebra. Even though these polytopes possess remarkable combinatorial and algebraic resemblance, they seem to be rarely unimodularly equivalent. In the present paper, we prove the following simple and elegant result: the order polytope and chain polytope for a poset are unimodularly equivallent if and only if that poset avoid the 5-element "X" shape subposet. We also explore a few equivalent statements of the main result.

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Raza, Hassan, Sakander Hayat, and Xiang-Feng Pan. "Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes." Symmetry 10, no. 12 (December 6, 2018): 727. http://dx.doi.org/10.3390/sym10120727.

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A convex polytope or simply polytope is the convex hull of a finite set of points in Euclidean space R d . Graphs of convex polytopes emerge from geometric structures of convex polytopes by preserving the adjacency-incidence relation between vertices. In this paper, we study the problem of binary locating-dominating number for the graphs of convex polytopes which are symmetric rotationally. We provide an integer linear programming (ILP) formulation for the binary locating-dominating problem of graphs. We have determined the exact values of the binary locating-dominating number for two infinite families of convex polytopes. The exact values of the binary locating-dominating number are obtained for two rotationally-symmetric convex polytopes families. Moreover, certain upper bounds are determined for other three infinite families of convex polytopes. By using the ILP formulation, we show tightness in the obtained upper bounds.

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DI ROCCO, SANDRA. "PROJECTIVE DUALITY OF TORIC MANIFOLDS AND DEFECT POLYTOPES." Proceedings of the London Mathematical Society 93, no. 1 (June 9, 2006): 85–104. http://dx.doi.org/10.1017/s0024611505015686.

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Non-singular toric embeddings with dual defect are classified. The associated polytopes, called defect polytopes, are proven to be the class of Delzant integral polytopes for which a combinatorial invariant vanishes. The structure of a defect polytope is described.

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Hibi, Takayuki, and Nan Li. "Cutting Convex Polytopes by Hyperplanes." Mathematics 7, no. 5 (April 26, 2019): 381. http://dx.doi.org/10.3390/math7050381.

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Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the original polytope are hereditary to its subpolytopes obtained by a cut. In this work, we devote our attention to all the separating hyperplanes for some given polytope (integral and convex) and study the existence and classification of such hyperplanes. We prove the existence of separating hyperplanes for the order and chain polytopes for any finite posets that are not a single chain, and prove there are no such hyperplanes for any Birkhoff polytopes. Moreover, we give a complete separating hyperplane classification for the unit cube and its subpolytopes obtained by one cut, together with some partial classification results for order and chain polytopes.

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Cunningham, Gabe, Mark Mixer, and Gordon Williams. "Reflexible covers of prisms." Contributions to Discrete Mathematics 19, no. 3 (September 23, 2024): 196–208. http://dx.doi.org/10.55016/ojs/cdm.v19i3.74917.

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The Tomotope provided the first well understood example of an abstract 4-polytope whose connection (monodromy) group was not a string C-group, and which also did not have a unique minimal regular cover. Conversely, we know that if the connection group of a polytope is a string C-group (if the polytope is C-connected), then the polytope will have a unique minimal regular cover. Since the discovery of the Tomotope, an active area of investigation has been determining which abstract $d$-polytopes are C-connected and the ways various constructions for abstract polytopes result in polytopes that do or do not possess unique minimal regular covers. In the current work we show that the prism over every abstract polyhedron is C-connected, or equivalently, that it has a unique minimal regular cover. We also describe a conjecture positing a general condition on the C-connectedness of prisms over polytopes that is independent of rank.

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9

Gubeladze, Joseph. "Affine-compact functors." Advances in Geometry 19, no. 4 (October 25, 2019): 487–504. http://dx.doi.org/10.1515/advgeom-2019-0004.

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Abstract Several well known polytopal constructions are examined from the functorial point of view. A naive analogy between the Billera–Sturmfels fiber polytope and the abelian kernel is disproved by an infinite explicit series of polytopes. A correct functorial formula is provided in terms of an affine-compact substitute of the abelian kernel. The dual cokernel object is almost always the natural affine projection. The Mond–Smith–van Straten space of sandwiched simplices, useful in stochastic factorizations, leads to a different kind of affine-compact functors and new challenges in polytope theory.

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Okay, Cihan, Ho Yiu Chung, and Selman Ipek. "Mermin polytopes quantum computation and foundations." Quantum Information & Computation 23, no. 9&10 (July 2023): 733–82. http://dx.doi.org/10.26421/qic23.9-10-2.

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Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\MP_\beta$ obtained from the Mermin scenario, parametrized by a function $\beta$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\MP_0$ and $\MP_1$ depending on the parity of $\beta$. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\MP_0$ turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. $\MP_1$ can be seen as a nonlocal toy version of $\Lambda$-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the $2$-qubit case, we provide a decomposition of the $\Lambda$-polytope using $\MP_1$, whose vertices are classified, and the nonsignaling polytope of the $(2,3,2)$ Bell scenario, whose vertices are well-known.

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Lee, Yonggyu, and Fu Liu. "Deformation cones of Tesler polytopes." Advances in Geometry 24, no. 2 (April 1, 2024): 209–27. http://dx.doi.org/10.1515/advgeom-2024-0003.

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Abstract For a ∈ $\begin{array}{} \displaystyle \mathbb{R}_{\geq 0}^{n} \end{array}$ , the Tesler polytope Tes n ( a ) is the set of upper triangular matrices with non-negative entries whose hook sum vector is a . We first give a different proof of the known fact that for every fixed a 0 ∈ $\begin{array}{} \displaystyle \mathbb{R}_{ \gt 0}^{n} \end{array}$ , all the Tesler polytopes Tes n ( a ) are deformations of Tes n ( a 0). We then calculate the deformation cone of Tes n ( a 0). In the process, we also show that any deformation of Tes n ( a 0) is a translation of a Tesler polytope. Lastly, we consider a larger family of polytopes called flow polytopes which contains the family of Tesler polytopes and chracterize the flow polytopes which are deformations of Tes n ( a 0).

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Matsuda, Kazunori, Hidefumi Ohsugi, and Kazuki Shibata. "Toric Rings and Ideals of Stable Set Polytopes." Mathematics 7, no. 7 (July 10, 2019): 613. http://dx.doi.org/10.3390/math7070613.

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In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their toric ideals. In particular, we give a criterion for the normality of the toric ring of the stable set polytope and a graph-theoretical characterization of the set of generators of the toric ideal of the stable set polytope for a graph of stability number two. As an application, we provide an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases.

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Kozachok, M. A. "Perfect Prismatoids and the Conjecture Concerning Face Numbers of Centrally Symmetric Polytopes." Modeling and Analysis of Information Systems 19, no. 6 (March 12, 2015): 137–47. http://dx.doi.org/10.18255/1818-1015-2012-6-137-147.

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In this paper we introduce and study a class of centrally symmetric polytopes – perfect prismatoids – and some its properties related to the famous conjecture concerning face numbers of centrally symmetric polytopes are proved. It is proved that any Hanner polytope is a perfect prismatoid and any perfect prismatoid is affine equivalent to some 0/1-polytope.

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EMIRIS, IOANNIS Z., VISSARION FISIKOPOULOS, CHRISTOS KONAXIS, and LUIS PEÑARANDA. "AN ORACLE-BASED, OUTPUT-SENSITIVE ALGORITHM FOR PROJECTIONS OF RESULTANT POLYTOPES." International Journal of Computational Geometry & Applications 23, no. 04n05 (August 2013): 397–423. http://dx.doi.org/10.1142/s0218195913600108.

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We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertex- and halfspace-representations of the polytope using an oracle producing resultant vertices in a given direction, thus avoiding walking on the polytope whose dimension is α - n - 1, where the input consists of α points in ℤn. Our approach is output-sensitive as it makes one oracle call per vertex and per facet. It extends to any polytope whose oracle-based definition is advantageous, such as the secondary and discriminant polytopes. Our publicly available implementation uses the experimental CGAL package triangulation. Our method computes 5-, 6- and 7- dimensional polytopes with 35K, 23K and 500 vertices, respectively, within 2hrs, and the Newton polytopes of many important surface equations encountered in geometric modeling in < 1sec, whereas the corresponding secondary polytopes are intractable. It is faster than tropical geometry software up to dimension 5 or 6. Hashing determinantal predicates accelerates execution up to 100 times. One variant computes inner and outer approximations with, respectively, 90% and 105% of the true volume, up to 25 times faster.

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Lalvani, Haresh. "Higher Dimensional Periodic Table Of Regular And Semi-Regular Polytopes." International Journal of Space Structures 11, no. 1-2 (April 1996): 155–71. http://dx.doi.org/10.1177/026635119601-222.

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This paper presents a higher-dimensional periodic table of regular and semi-regular n-dimensional polytopes. For regular n-dimensional polytopes, designated by their Schlafli symbol {p,q,r,…u,v,w}, the table is an (n-1)-dimensional hypercubic lattice in which each polytope occupies a different vertex of the lattice. The values of p,q,r,…u,v,w also establish the corresponding n-dimensional Cartesian co-ordinates (p,q,r,…u,v,w) of their respective positions in the hypercubic lattice. The table is exhaustive and includes all known regular polytopes in Euclidean, spherical and hyperbolic spaces, in addition to others candidate polytopes which do not appear in the literature. For n-dimensional semi-regular polytopes, each vertex of this hypercubic lattice branches into analogous n-dimensional cubes, where each n-cube encompasses a family with a distinct semi-regular polytope occupying each vertex of each n-cube. The semi-regular polytopes are obtained by varying the location of a vertex within the fundamental region of the polytope. Continuous transformations within each family are a natural fallout of this variable vertex location. Extensions of this method to less regular space structures and to derivation of architectural form are in progress and provide a way to develop an integrated index for space structures. Besides the economy in computational processing of space structures, integrated indices based on unified morphologies are essential for establishing a meta-structural knowledge base for architecture.

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Gaubert, Stéphane, and Marie MacCaig. "Approximating the volume of tropical polytopes is difficult." International Journal of Algebra and Computation 29, no. 02 (March 2019): 357–89. http://dx.doi.org/10.1142/s0218196719500061.

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We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor [Formula: see text] for the volume of a tropical polytope given by [Formula: see text] for the volume of a tropical polytope given by [Formula: see text] vertices in a space of dimension [Formula: see text], unless P[Formula: see text]NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. It follows that approximating these values for tropical polytopes is more difficult than for classical polytopes. Our proofs use a reduction from the problem of calculating the tropical rank.

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Chimani, Markus, Martina Juhnke-Kubitzke, and Alexander Nover. "On the bond polytope." Advances in Geometry 23, no. 4 (October 1, 2023): 461–80. http://dx.doi.org/10.1515/advgeom-2023-0014.

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Abstract While the maximum cut problem and its corresponding polytope has received a lot of attention inliterature, comparably little is known about the natural closely related variant maximum bond. Here, given a graph G = (V, E), we ask for a maximum cut δ(S) ⊆ E with S ⊆ V under the restriction that both G[S] as well as G[V \ S] are connected. Observe that both the maximum cut and the maximum bond can be seen as inverse problems to the traditional minimum cut, as there, the connectivity arises naturally in optimal solutions. The bond polytope is the convex hull of all incidence vectors of bonds. Similar to the connection of the corresponding optimization problems, the bond polytope is closely related to the cut polytope. While the latter has been intensively studied, there are no results on bond polytopes. We start a structural study of the latter, which additionally allows us to deduce algorithmic consequences. We investigate the relation between cut- and bond polytopes and the additional intricacies that arise when requiring connectivity in the solutions. We study the effect of graph modifications on bond polytopes and their facets, akin to what has been spearheaded for cut polytopes by Barahona, Grötschel and Mahjoub [4; 3] and Deza and Laurant [17; 15; 16]. Moreover, we study facet-defining inequalities arising from edges and cycles for bond polytopes. In particular, these yield a complete linear description of bond polytopes of cycles and 3-connected planar (K 5 − e)-minor free graphs. Finally, we present a reduction of the maximum bond problem on arbitrary graphs to the maximum bond problem on 3-connected graphs. This yields a linear time algorithm for maximum bond on (K 5 − e)-minor free graphs.

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Hyeon, Donghoon, and Jaekwang Kim. "A State Polytope Decomposition Formula." Proceedings of the Edinburgh Mathematical Society 59, no. 3 (December 14, 2015): 759–76. http://dx.doi.org/10.1017/s0013091515000401.

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AbstractWe give a decomposition formula for computing the state polytope of a reducible variety in terms of the state polytopes of its components: if a polarized projective variety X is a chain of subvarieties Xi satisfying some further conditions, then the state polytope of X is the Minkowski sum of the state polytopes of Xi translated by a vector τ, which can be readily computed from the ideal of Xi. The decomposition is in the strongest sense in that the vertices of the state polytope of X are precisely the sum of vertices of the state polytopes of Xi translated by τ. We also give a similar decomposition formula for the Hilbert–Mumford index of the Hilbert points of X. We give a few examples of the state polytope and the Hilbert–Mumford index computation of reducible curves, which are interesting in the context of the log minimal model program for the moduli space of stable curves.

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Beck, Matthias, Danai Deligeorgaki, Max Hlavacek, and Jerónimo Valencia-Porras. "Inequalities for f ^*-vectors of lattice polytopes." Advances in Geometry 24, no. 2 (April 1, 2024): 141–50. http://dx.doi.org/10.1515/advgeom-2024-0002.

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Abstract The Ehrhart polynomial ehr P (n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f *-vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of ehr P (n) with respect to the binomial coefficient basis $\begin{array}{} \bigl\{\binom{n-1}{0},\binom{n-1}{1},\dots,\binom{n-1}{d}\bigr\}, \end{array}$ where d = dim P. Similarly to h/h *-vectors, the f *-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f *-vectors of lattice polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f *-coefficients increases and the last quarter decreases. Even though f *-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h *-vector, there is a polytope with the same h *-vector whose f *-vector is unimodal.

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Matczak, K., A. Mućka, and A. B. Romanowska. "Duality for dyadic intervals." International Journal of Algebra and Computation 29, no. 01 (February 2019): 41–60. http://dx.doi.org/10.1142/s0218196718500625.

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In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of (real) polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. This paper is a first step in finding a duality for dyadic polytopes, analogues of real convex polytopes, but defined over the ring [Formula: see text] of dyadic rational numbers instead of the ring of reals. A dyadic [Formula: see text]-dimensional polytope is the intersection with the dyadic space [Formula: see text] of an [Formula: see text]-dimensional real polytope whose vertices lie in the dyadic space. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polytopes carry the structure of a commutative, entropic and idempotent groupoid under the operation of arithmetic mean. Such dyadic polytopes do not preserve all properties of real polytopes. In particular, there are infinitely many (pairwise non-isomorphic) dyadic intervals. We first show that finitely generated subgroupoids of the groupoid [Formula: see text] are all isomorphic to dyadic intervals. Then, we describe a duality for the class of dyadic intervals. The duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the square of the dyadic unit interval with additional constant operations. A second paper deals with a duality for dyadic triangles.

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Rasmussen, Jørgen, and Mark A. Walton. "Demazure formula for A_n Weyl polytope sums." Journal of Mathematical Physics 62, no. 10 (October 1, 2021): 101702. http://dx.doi.org/10.1063/5.0058465.

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The weights of finite-dimensional representations of simple Lie algebras are naturally associated with Weyl polytopes. Representation characters decompose into multiplicity-free sums over the weights in Weyl polytopes. The Brion formula for these Weyl polytope sums is remarkably similar to the Weyl character formula. Moreover, the same Lie characters are also expressible as Demazure character formulas. This motivates a search for new expressions for Weyl polytope sums, and we prove such a formula involving Demazure operators. It applies to the Weyl polytope sums of the simple Lie algebras A n for all dominant integrable highest weights and all ranks n.

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Lee, Jae-Hyouk. "Gosset Polytopes in Picard Groups of del Pezzo Surfaces." Canadian Journal of Mathematics 64, no. 1 (February 1, 2012): 123–50. http://dx.doi.org/10.4153/cjm-2011-063-6.

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Abstract In this article, we study the correspondence between the geometry of del Pezzo surfaces Sr and the geometry of the r-dimensional Gosset polytopes (r − 4)21. We construct Gosset polytopes (r −4)21 in Pic Sr ꕕ ℚ whose vertices are lines, and we identify divisor classes in Pic Sr corresponding to (a − 1)-simplexes (a ≤ r), (r − 1)-simplexes and (r − 1)-crosspolytopes of the polytope (r − 4)21. Then we explain how these classes correspond to skew a-lines(a ≤ r), exceptional systems, and rulings, respectively.As an application, we work on the monoidal transform for lines to study the local geometry of the polytope (r−4)21. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes 321 and 421, respectively.

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Ziegler, Günter M. "Additive structures on f-vector sets of polytopes." Advances in Geometry 20, no. 2 (April 28, 2020): 217–31. http://dx.doi.org/10.1515/advgeom-2018-0025.

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AbstractWe show that the f-vector sets of d-polytopes have non-trivial additive structure: They span affine lattices and are embedded in monoids that we describe explicitly. Moreover, for many large subclasses, such as the simple polytopes, or the simplicial polytopes, there are monoid structures on the set of f-vectors by themselves: “addition of f-vectors minus the f-vector of the d-simplex” always yields a new f-vector. For general 4-polytopes, we show that the modified addition operation does not always produce an f-vector, but that the result is always close to an f-vector. In this sense, the set of f-vectors of all 4-polytopes forms an “approximate affine semigroup”. The proof relies on the fact for d = 4 every d-polytope, or its dual, has a “small facet”. This fails for d > 4.We also describe a two further modified addition operations on f-vectors that can be geometrically realized by glueing corresponding polytopes. The second one of these may yield a semigroup structure on the f-vector set of all 4-polytopes.

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HILLE, LUTZ, and HARALD SKARKE. "REFLEXIVE POLYTOPES IN DIMENSION 2 AND CERTAIN RELATIONS IN SL2(ℤ)." Journal of Algebra and Its Applications 01, no. 02 (June 2002): 159–73. http://dx.doi.org/10.1142/s0219498802000124.

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It is well known that there are 16 two-dimensional reflexive polytopes up to lattice isomorphism. One can check directly from the list that the number of lattice points on the boundary of such a polytope plus the number of lattice points on the boundary of the dual polytope is always 12. It turns out that two-dimensional reflexive polytopes correspond to certain relations of two generators A and B of SL 2(ℤ) of length 12. We generalize this correspondence to reflexive configurations with winding number w and relations of length 12w.

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Maksimenko, Aleksandr N. "On a family of 0/1-polytopes with an NP-complete criterion for vertex nonadjacency relation." Discrete Mathematics and Applications 29, no. 1 (February 25, 2019): 7–14. http://dx.doi.org/10.1515/dma-2019-0002.

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Abstract In 1995 T. Matsui considered a special family of 0/1-polytopes with an NP-complete criterion for vertex nonadjacency relation. In 2012 the author demonstrated that all polytopes of this family appear as faces of polytopes associated with the following NP-complete problems: the travelling salesman problem, the 3-satisfiability problem, the knapsack problem, the set covering problem, the partial ordering problem, the cube subgraph problem, and some others. Here it is shown that none of the polytopes of the aforementioned special family (with the exception of the one-dimensional segment) can appear as a face in a polytope associated with the problem of the maximum independent set, the set packing problem, the set partitioning problem, and the problem of 3-assignments.

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Kim, Jin Hong. "On the first two entries of the f-vectors of 6-polytopes." Contributions to Discrete Mathematics 15, no. 1 (May 11, 2020): 90–101. http://dx.doi.org/10.55016/ojs/cdm.v15i1.62385.

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In 1906, Steinitz gave a complete characterization of the first two entries of the $f$-vectors of $3$-polytopes, while Grünbaum obtained a similar result for $4$-polytopes in his well-known book published in 1967. Recently, Kusunoki and Murai and independently Pineda-Villavicencio, Ugon, and Yost completely determined the first two entries of the $f$-vectors of $5$-polytopes. This paper can be regarded as a continuation of their works for $6$-polytopes. To be more precise, let $k$ denote the number of vertices of a $6$-polytope. The aim of this paper is to show that, when the number of edges is greater than or equal to $\frac{7}{2}(k-1)$ and $k\ge 14$, we can completely characterize the first two entries of the $f$-vectors of $6$-polytopes. As a consequence, for $7\le k\le 15$ we also give a complete characterization of the first two entries of the $f$-vectors of $6$-polytopes except for three cases $(12, 39)$, $(13, 43)$, and $(15, 47)$.

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Burger, Thomas, Peter Gritzmann, and Victor Klee. "Polytope Projection and Projection Polytopes." American Mathematical Monthly 103, no. 9 (November 1996): 742. http://dx.doi.org/10.2307/2974444.

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Burger, Thomas, Peter Gritzmann, and Victor Klee. "Polytope Projection and Projection Polytopes." American Mathematical Monthly 103, no. 9 (November 1996): 742–55. http://dx.doi.org/10.1080/00029890.1996.12004814.

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Hayat, Sakander, Muhammad Yasir Hayat Malik, Ali Ahmad, Suliman Khan, Faisal Yousafzai, and Roslan Hasni. "On Hamilton-Connectivity and Detour Index of Certain Families of Convex Polytopes." Mathematical Problems in Engineering 2021 (July 17, 2021): 1–18. http://dx.doi.org/10.1155/2021/5553216.

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A convex polytope is the convex hull of a finite set of points in the Euclidean space ℝ n . By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. The detour index is defined to be the sum of the lengths of longest distances, i.e., detours between vertices in a graph. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering, whereas the detour index has important applications in chemistry. Checking whether a graph is Hamilton-connected and computing the detour index of an arbitrary graph are both NP-complete problems. In this paper, we study these problems simultaneously for certain families of convex polytopes. We construct two infinite families of Hamilton-connected convex polytopes. Hamilton-connectivity is shown by constructing Hamiltonian paths between any pair of vertices. We then use the Hamilton-connectivity to compute the detour index of these families. A family of non-Hamilton-connected convex polytopes has also been constructed to show that not all convex polytope families are Hamilton-connected.

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D’Azevedo, Antonio Breda, Gareth A. Jones, and Egon Schulte. "Constructions of Chiral Polytopes of Small Rank." Canadian Journal of Mathematics 63, no. 6 (December 1, 2011): 1254–83. http://dx.doi.org/10.4153/cjm-2011-033-4.

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AbstractAn abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. This paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4, and 5.

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CONNOR, THOMAS, DIMITRI LEEMANS, and MARK MIXER. "ABSTRACT REGULAR POLYTOPES FOR THE O'NAN GROUP." International Journal of Algebra and Computation 24, no. 01 (February 2014): 59–68. http://dx.doi.org/10.1142/s0218196714500052.

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In this paper, we consider how the O'Nan sporadic simple group acts as the automorphism group of an abstract regular polytope. In particular, we prove that there is no regular polytope of rank at least five with automorphism group isomorphic to O′N. Moreover, we classify all rank four regular polytopes having O′N as their automorphism group.

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Deza, Antoine, Jean-Baptiste Hiriart-Urruty, and Lionel Pournin. "Polytopal balls arising in optimization." Contributions to Discrete Mathematics 16, no. 3 (December 31, 2021): 125–38. http://dx.doi.org/10.55016/ojs/cdm.v16i3.71526.

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We study a family of polytopes and their duals, that appear in various optimization problems as the unit balls for certain norms. These two families interpolate between the hypercube, the unit ball for the $\infty$-norm, and its dual cross-polytope, the unit ball for the $1$-norm. We give combinatorial and geometric properties of both families of polytopes such as their $f$-vector, their volume, and the volume of their boundary.

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Krause, Oswin, Bertram Brovang, Torbjørn Rasmussen, Anasua Chatterjee, and Ferdinand Kuemmeth. "Estimation of Convex Polytopes for Automatic Discovery of Charge State Transitions in Quantum Dot Arrays." Electronics 11, no. 15 (July 27, 2022): 2327. http://dx.doi.org/10.3390/electronics11152327.

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In spin based quantum dot arrays, material or fabrication imprecisions affect the behaviour of the device, which must be taken into account when controlling it. This requires measuring the shape of specific convex polytopes. We present an algorithm that automatically discovers count, shape and size of the facets of a convex polytope from measurements by alternating a phase of model-fitting with a phase of querying new measurements, based on the fitted model. We evaluate the algorithm on simulated polytopes and devices, as well as a real 2 × 2 spin qubit array. Results show that we can reliably find the facets of the convex polytopes, including small facets with sizes on the order of the measurement precision.

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Ahn, Hee-Kap, Seung Joon Lee, and Sang Duk Yoon. "Stacking Monotone Polytopes." Symmetry 16, no. 9 (September 23, 2024): 1246. http://dx.doi.org/10.3390/sym16091246.

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This paper addresses the problem of computing the optimal stacking of two monotone polytopes P and Q in Rd. A monotone polytope in Rd is defined as a polytope whose intersection with any line parallel to the last coordinate axis xd is connected, and the stacking of P and Q is defined as a translation of Q, such that “Q touches P from above”. To evaluate the stack, we use three different scoring criteria: (1) the height of the stack, (2) the maximum pointwise distance along the xd-axis, and (3) the volume between P and Q. We propose exact algorithms to compute the optimal stacking for each scoring criterion.

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Andrews, Lawrence C., and Herbert J. Bernstein. "The geometry of Niggli reduction:BGAOL–embedding Niggli reduction and analysis of boundaries." Journal of Applied Crystallography 47, no. 1 (January 30, 2014): 346–59. http://dx.doi.org/10.1107/s1600576713031002.

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Niggli reduction can be viewed as a series of operations in a six-dimensional space derived from the metric tensor. An implicit embedding of the space of Niggli-reduced cells in a higher-dimensional space to facilitate calculation of distances between cells is described. This distance metric is used to create a program,BGAOL, for Bravais lattice determination. Results fromBGAOLare compared with results from other metric based Bravais lattice determination algorithms. This embedding depends on understanding the boundary polytopes of the Niggli-reduced coneNin the six-dimensional spaceG6. This article describes an investigation of the boundary polytopes of the Niggli-reduced coneNin the six-dimensional spaceG6by algebraic analysis and organized random probing of regions near one-, two-, three-, four-, five-, six-, seven- and eightfold boundary polytope intersections. The discussion of valid boundary polytopes is limited to those avoiding the mathematically interesting but crystallographically impossible cases of zero-length cell edges. Combinations of boundary polytopes without a valid intersection in the closure of the Niggli cone or with an intersection that would force a cell edge to zero or without neighboring probe points are eliminated. In all, 216 boundary polytopes are found. There are 15 five-dimensional boundary polytopes of the fullG6Niggli coneN.

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Hoessly, Linard. "Politopos en filogenética." Tequio 4, no. 11 (January 27, 2021): 27–40. http://dx.doi.org/10.53331/teq.v4i11.0649.

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We introduce mathematical notions used in phylogenetics and three sorts of phylogenetics polytopes. The Tight span and the Lipschitz polytope are both associated to finite metric spaces and can be connected to distance-preserving embeddings, while the balanced minimum evolution (BME) polytope is associated to natural numbers.

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Chen, Zhi, Zelin Zhu, Jiawei Li, Lizhen Yang, and Lei Cao. "Extreme Points of Certain Transportation Polytopes with Fixed Total Sums." Electronic Journal of Linear Algebra 37 (April 5, 2021): 256–71. http://dx.doi.org/10.13001/ela.2021.5141.

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Transportation matrices are $m\times n$ nonnegative matrices with given row sum vector $R$ and column sum vector $S$. All such matrices form the convex polytope $\mathcal{U}(R,S)$ which is called a transportation polytope and its extreme points have been classified. In this article, we consider a new class of convex polytopes $\Delta(\bar{R},\bar{S},\sigma)$ consisting of certain transportation polytopes satisfying that the sum of all elements is $\sigma$, and the row and column sum vectors are dominated componentwise by the given positive vectors $\bar{R}$ and $\bar{S}$, respectively. We characterize the extreme points of $\Delta(\bar{R},\bar{S},\sigma)$. Moreover, we give the minimal term rank and maximal permanent of $\Delta(\bar{R},\bar{S},\sigma)$.

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Dutour Sikirić, Mathieu. "The Seven Dimensional Perfect Delaunay Polytopes and Delaunay Simplices." Canadian Journal of Mathematics 69, no. 5 (October 1, 2017): 1143–68. http://dx.doi.org/10.4153/cjm-2016-013-7.

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AbstractFor a lattice L of ℝn, a sphere S(c, r) of center c and radius r is called empty if for any v ∈ L we have. Then the set S(c, r) ∩ L is the vertex set of a Delaunay polytope P = conv(S(c, r) ∩ L). A Delaunay polytope is called perfect if any aõne transformation ø such that ø(P) is a Delaunay polytope is necessarily an isometry of the space composed with an homothety.Perfect Delaunay polytopes are remarkable structures that exist only if n = 1 or n ≥ 6, and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension seven, which allows us to find that there are only two perfect Delaunay polytopes: 321, which is a Delaunay polytope in the root lattice E7, and the Erdahl Rybnikov polytope.We then use this classification in order to get the list of all types of Delaunay simplices in dimension seven and found that there are eleven types.

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Melikhova, E. "On the number of faces of the Gelfand–Zetlin polytope." St. Petersburg Mathematical Journal 33, no. 3 (May 5, 2022): 553–68. http://dx.doi.org/10.1090/spmj/1714.

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The combinatorics of the Gelfand–Zetlin polytope is studied. Geometric properties of a linear projection of this polytope onto a cube are employed to derive a recurrence relation for the f f -polynomial of the polytope. This recurrence relation is applied to finding the f f -polynomials and h h -polynomials for one-parameter families of Gelfand–Zetlin polytopes of simplest types.

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40

O'Cinneide, Colm Art. "Phase-type distributions and invariant polytopes." Advances in Applied Probability 23, no. 3 (September 1991): 515–35. http://dx.doi.org/10.2307/1427620.

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The notion of an invariant polytope played a central role in the proof of the characterization of phase-type distributions. The purpose of this paper is to develop invariant polytope techniques further. We derive lower bounds on the number of states needed to represent a phase-type distribution based on poles of its Laplace–Stieltjes transform. We prove that every phase-type distribution whose transform has only real poles has a bidiagonal representation. We close with three short applications of the invariant polytope idea. Taken together, the results of this paper show that invariant polytopes provide a natural approach to many questions about phase-type distributions.

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O'Cinneide, Colm Art. "Phase-type distributions and invariant polytopes." Advances in Applied Probability 23, no. 03 (September 1991): 515–35. http://dx.doi.org/10.1017/s0001867800023715.

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The notion of an invariant polytope played a central role in the proof of the characterization of phase-type distributions. The purpose of this paper is to develop invariant polytope techniques further. We derive lower bounds on the number of states needed to represent a phase-type distribution based on poles of its Laplace–Stieltjes transform. We prove that every phase-type distribution whose transform has only real poles has a bidiagonal representation. We close with three short applications of the invariant polytope idea. Taken together, the results of this paper show that invariant polytopes provide a natural approach to many questions about phase-type distributions.

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Ardila, Federico, Alex Fink, and Felipe Rincón. "Valuations for Matroid Polytope Subdivisions." Canadian Journal of Mathematics 62, no. 6 (December 14, 2010): 1228–45. http://dx.doi.org/10.4153/cjm-2010-064-9.

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43

Chakrabarti, Darshan, Gabriele Farina, and Christian Kroer. "Efficient Learning in Polyhedral Games via Best-Response Oracles." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 9 (March 24, 2024): 9564–72. http://dx.doi.org/10.1609/aaai.v38i9.28812.

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We study online learning and equilibrium computation in games with polyhedral decision sets, a property shared by normal-form games (NFGs) and extensive-form games (EFGs), when the learning agent is restricted to utilizing a best-response oracle. We show how to achieve constant regret in zero-sum games and O(T^0.25) regret in general-sum games while using only O(log t) best-response queries at a given iteration t, thus improving over the best prior result, which required O(T) queries per iteration. Moreover, our framework yields the first last-iterate convergence guarantees for self-play with best-response oracles in zero-sum games. This convergence occurs at a linear rate, though with a condition-number dependence. We go on to show a O(T^(-0.5)) best-iterate convergence rate without such a dependence. Our results build on linear-rate convergence results for variants of the Frank-Wolfe (FW) algorithm for strongly convex and smooth minimization problems over polyhedral domains. These FW results depend on a condition number of the polytope, known as facial distance. In order to enable application to settings such as EFGs, we show two broad new results: 1) the facial distance for polytopes in standard form is at least γ/k where γ is the minimum value of a nonzero coordinate of a vertex of the polytope and k≤n is the number of tight inequality constraints in the optimal face, and 2) the facial distance for polytopes of the form Ax=b, Cx≤d, x≥0 where x∈R^n, C≥0 is a nonzero integral matrix, and d≥0, is at least 1/(c√n), where c is the infinity norm of C. This yields the first such results for several problems such as sequence-form polytopes, flow polytopes, and matching polytopes.

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BATTAGLIA, FIAMMETTA. "GEOMETRIC SPACES FROM ARBITRARY CONVEX POLYTOPES." International Journal of Mathematics 23, no. 01 (January 2012): 1250013. http://dx.doi.org/10.1142/s0129167x11007562.

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We associate a geometric space to an arbitrary convex polytope. The spaces that we obtain are endowed with a natural stratification and perfectly mimic the features of toric varieties associated to rational convex polytopes. Our construction generalizes the classic constructions of Cox and Delzant of toric varieties.

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Lawless, Connor, Jayant Kalagnanam, Lam M. Nguyen, Dzung Phan, and Chandra Reddy. "Interpretable Clustering via Multi-Polytope Machines." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 7 (June 28, 2022): 7309–16. http://dx.doi.org/10.1609/aaai.v36i7.20693.

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Clustering is a popular unsupervised learning tool often used to discover groups within a larger population such as customer segments, or patient subtypes. However, despite its use as a tool for subgroup discovery and description few state-of-the-art algorithms provide any rationale or description behind the clusters found. We propose a novel approach for interpretable clustering that both clusters data points and constructs polytopes around the discovered clusters to explain them. Our framework allows for additional constraints on the polytopes including ensuring that the hyperplanes constructing the polytope are axis-parallel or sparse with integer coefficients. We formulate the problem of constructing clusters via polytopes as a Mixed-Integer Non-Linear Program (MINLP). To solve our formulation we propose a two phase approach where we first initialize clusters and polytopes using alternating minimization, and then use coordinate descent to boost clustering performance. We benchmark our approach on a suite of synthetic and real world clustering problems, where our algorithm outperforms state of the art interpretable and non-interpretable clustering algorithms.

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Tsuda, Ren, and Takanori Fujiwara. "Oscillating 4-Polytopal Universe in Regge Calculus." Progress of Theoretical and Experimental Physics, June 29, 2021. http://dx.doi.org/10.1093/ptep/ptab079.

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Abstract The discretized closed Friedmann-Lemaître-Robertson-Walker (FLRW) universe with positive cosmological constant is investigated by Regge calculus. According to the Collins-Williams formalism, a hyperspherical Cauchy surface is replaced with regular 4-polytopes. Numerical solutions to the Regge equations approximate well to the continuum solution during the era of small edge length. Unlike the expanding polyhedral universe in three dimensions the 4-polytopal universes repeat expansions and contractions. To go beyond the approximation by regular 4-polytopes we introduce pseudo-regular 4-polytopes by averaging dihedral angles of the tessellated regular 600-cell. Degree of precision of the tessellation is called the frequency. Regge equations for the pseudo-regular 4-polytope have simple and unique expression for any frequency. In the infinite frequency limit, the pseudo-regular 4-polytope model approaches the continuum FLRW universe.

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47

Kim, Sangwook. "Flag enumerations of matroid base polytopes." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AJ,..., Proceedings (January 1, 2008). http://dx.doi.org/10.46298/dmtcs.3640.

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International audience In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the $\textbf{cd}$-index of a polytope can be expressed when a polytope is cut by a hyperplane, and apply these to the $\textbf{cd}$-index of a matroid base polytope of a rank $2$ matroid. Dans cet article, nous étudions les structures de drapeau de polytopes de base de matroïde. Nous décrivons des faces de polytopes de base de matroïde en terme des données de matroïde, et donnons des conditions pour les divisions de hyperplane de polytopes de base de matroïde. Nous montrons également comment le $\mathbf{cd}$-index d’un polytope peut être exprimé quand un polytope est coupé par un hyperplane, et appliquons ceux-ci au $\mathbf{cd}$-index d’un polytope de base de matroïde d’un rang $2$ matroïde.

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Escobar, Laura, and Karola Mészáros. "Toric matrix Schubert varieties and root polytopes (extended abstract)." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6405.

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International audience Start with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety Xπ. We characterize when the ideal defining Xπ is toric (with respect to a 2n − 1-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials. Subword complexes were introduced by Knutson and Miller in 2004, who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.

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Werner, Annette, and Josephine Yu. "Symmetric Alcoved Polytopes." Electronic Journal of Combinatorics 21, no. 1 (January 24, 2014). http://dx.doi.org/10.37236/3646.

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Generalized alcoved polytopes are polytopes whose facet normals are roots in a given root system. We call a set of points in an alcoved polytope a generating set if there does not exist a strictly smaller alcoved polytope containing it. The type $A$ alcoved polytopes are precisely the tropical polytopes that are also convex in the usual sense. In this case the tropical generators form a generating set. We show that for any root system other than $F_4$, every alcoved polytope invariant under the natural Weyl group action has a generating set of cardinality equal to the Coxeter number of the root system.

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Freij-Hollanti, Ragnar, and Teemu Lundström. "$f$-vector inequalities for order and chain polytopes." MATHEMATICA SCANDINAVICA 130, no. 3 (November 4, 2024). http://dx.doi.org/10.7146/math.scand.a-143995.

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The order and chain polytopes are two $0/1$-polytopes constructed from a finite poset. In this paper, we study the $f$-vectors of these polytopes. We investigate how the order and chain polytopes behave under disjoint unions and ordinal sums of posets, and how the $f$-vectors of these polytopes are expressed in terms of $f$-vectors of smaller polytopes. Our focus is on comparing the $f$-vectors of the order and chain polytope built from the same poset. In our main theorem we prove that for a family of posets built inductively by taking disjoint unions and ordinal sums of posets, for any poset $\mathcal {P}$ in this family the $f$-vector of the order polytope of $\mathcal {P}$ is component-wise at most the $f$-vector of the chain polytope of $\mathcal {P}$.

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Journal articles on the topic 'Polytopes'

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