Polyhedral Complex Research Articles

Holes and dependences in an ordered complex

We use the canonical bases produced by the tri-partition algorithm in (Edelsbrunner and Ölsböck, 2018) to open and close holes in a polyhedral complex, K. In a concrete application, we consider the Delaunay mosaic of a finite set, we let K be an Alpha complex, and we use the persistence diagram of the distance function to guide the hole opening and closing operations. The dependences between the holes define a partial order on the cells in K that characterizes what can and what cannot be constructed using the operations. The relations in this partial order reveal structural information about the underlying filtration of complexes beyond what is expressed by the persistence diagram.

Discrete volumes of lattice polyhedra via vector analysis

Pick’s theorem relates the number of lattice points to the area for a lattice polygon. Diaz and Robins gave a proof of Pick’s theorem by using the Weierstrass $$\wp $$ -function and complex analysis. As an analogue to lattice convex polyhedra, Reeve’s theorem is known as a solid version of Pick’s theorem. In this paper, we study counting lattice points on a lattice polyhedron by using vector analysis, and we extend Reeve’s theorem to nonconvex polyhedral complexes.

Consistently Oriented Dart-based 3D Modelling by Means of Geometric Algebra and Combinatorial Maps

The modelling of real world objects is not a straightforward subject. There are many different schemes; constructive solid geometry (CSG), cell decomposition, boundary representation, etcetera. Obviously, somehow, any scheme will be related to any other since they have a common goal. The paper shows how to model general polyhedra as an unordered discrete and finite set of geometric numbers of a projective Clifford Algebra or Geometric Algebra (GA). Clearly, not any randomly generated finite set of geometric numbers will have the structure of an object, this set must have some well defined properties. The topological properties extracted from this set are mapped to a boundary representation scheme based on a type of combinatorial map called generalised map or n-gmap. The n-gmaps have different types of orbits (in the mathematical sense) to which an attribute can be attached. When the attribute has a geometrical meaning, it is said that it is the geometrical embedding of the n-gmap. In this way the n-gmap holds explicitly the topology or structure already defined by the discrete geometry. In our proposal, each single element of a n-gmap is consistently embedded into a geometrical number also known as multi-vector. The scheme has been implemented by modifying an open source code [46] of n-gmaps. This representation has interesting properties. GA and n-gmaps complement and reinforce each other. For instance; it improves the robustness when computing the structure from the geometrical information. It is capable of computing lengths, areas and volumes of any polyhedral complex (with or without holes) using the orbits of the n-gmap (some examples are given). Finally the paper gives hints about other potentialities.

Power Diagram Detection with Applications to Information Elicitation

Power diagrams, a type of weighted Voronoi diagram, have many applications throughout operations research. We study the problem of power diagram detection: determining whether a given finite partition of $${\mathbb {R}}^d$$ takes the form of a power diagram. This detection problem is particularly prevalent in the field of information elicitation, where one wishes to design contracts to incentivize self-minded agents to provide honest information. We devise a simple linear program to decide whether a polyhedral cell complex can be described as a power diagram. For positive instances, a representation of the cell complex as a power diagram is returned. Further, we discuss applications to property elicitation, peer prediction, and mechanism design, where this question arises. Our model can efficiently decide the question for complexes of $${\mathbb {R}}^d$$ or of a convex subset thereof. The approach is based on the use of an alternative representation of power diagrams and invariance of a power diagram under uniform scaling of the parameters in this representation.

Piecewise polynomials with different smoothness degrees on polyhedral complexes

For a given d-dimensional polyhedral complex Δ and a given degree k, we consider the vector space of piecewise polynomial functions on Δ of degree at most k with a different smoothness condition on each pair of adjacent d-faces of Δ. This is a finite dimensional vector space. The fundamental problem in Approximation Theory is to compute the dimension of this vector space. It is known that the dimension is given by a polynomial for sufficiently large k via commutative algebra. By using the technique of McDonald and Schenck [3] and extending their result to a plane polyhedral complex Δ with varying smoothness conditions, we determine this polynomial. This gives a complete answer for the dimension. At the end we discuss some examples through this technique.

Homomorphism complexes and [formula omitted]-cores

For any fixed graph G, we prove that the topological connectivity of the graph homomorphism complex Hom(G,Km) is at least m−D(G)−2, where D(G)=maxH⊆Gδ(H), for δ(H) the minimum degree of a vertex in a subgraph H. This generalizes a theorem of C̆ukić and Kozlov, in which the maximum degree Δ(G) was used in place of D(G), and provides a high-dimensional analogue of the graph theoretic bound for chromatic number, χ(G)≤D(G)+1, as χ(G)=min{m:Hom(G,Km)≠∅}. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom(G(n,p),Km) when p=c∕n for a fixed constant c>0.

Weakly well-composed cell complexes over nD pictures

In previous work we proposed a combinatorial algorithm to “locally repair” the cubical complex Q(I) that is canonically associated with a given 3D picture I. The algorithm constructs a 3D polyhedral complex P(I) which is homotopy equivalent to Q(I) and whose boundary surface is a 2D manifold. A polyhedral complex satisfying these properties is called well-composed. In the present paper we extend these results to higher dimensions. We prove that for a given n-dimensional picture the obtained cell complex is well-composed in a weaker sense but is still homotopy equivalent to the initial cubical complex.

Amoeba-Shaped Polyhedral Complex of an Algebraic Hypersurface

Given a complex algebraic hypersurface H, we introduce a subset of the Newton polytope of the defining polynomial for H which is a polyhedral complex and enjoys the key topological and combinatorial properties of the amoeba of H for a large class of hypersurfaces. We provide an explicit formula for this polyhedral complex in the case when the spine of the amoeba is dual to a triangulation of the Newton polytope of the defining polynomial. In particular, this yields a description of the polyhedral complex when the hypersurface is optimal (Forsberg et al. in Adv Math 151:45–70, 2000). We conjecture that a polyhedral complex with these properties exists in general.

Warmth and edge spaces of graphs

In this paper we study a pair of numerical parameters associated to a graph G. On the one hand, we consider the (topological) connectivity of the polyhedral complex Hom(K2,G), a space of homomorphisms from a edge K2 into G. This approach dates back to the neighborhood complexes introduced by Lovász in his proof of the Kneser conjecture. In another direction we study the warmth ζ(G) of a graph G, a parameter introduced by Brightwell and Winkler based on the asymptotic behavior of d-branching walks in G and inspired by constructions in statistical physics. Both the warmth of G and the connectivity of Hom(K2,G) provide lower bounds on the chromatic number of G.Here we seek to relate these two constructions, and in particular we provide evidence for the conjecture that the warmth of a graph G is always less than three plus the connectivity of Hom(K2,G). We succeed in establishing a first nontrivial case of the conjecture, by showing that ζ(G)≤3 if Hom(K2,G) has an infinite first homology group. We also calculate warmth for a family of ‘twisted toroidal’ graphs that are important extremal examples in the context of Hom complexes. Finally we show that ζ(G)≤n−1 if a graph G does not have the complete bipartite graph Ka,b for a+b=n. This provides an analogue for a similar result in the context of Hom complexes.

Repairing 3D binary images using the BCC grid with a 4-valued combinatorial coordinate system

A 3D binary image I is called well-composed if the set of points in the topological boundary of the cubes in I is a 2-manifold. Repairing a 3D binary image is a process which produces a well composed image (or a polyhedral complex) from the non-well-composed image I.We propose here to repair 3D images by associating the Body-Centered Cubic grid (BCC grid) to the cubical grid. The obtained polyhedral complex is well composed, since two voxels in the BCC grid either share an entire face or are disjoint. We show that the obtained complex is homotopy equivalent to the cubical complex naturally associated with the image I.To efficiently encode and manipulate the BCC grid, we present an integer 4-valued combinatorial coordinate system that addresses cells of all dimensions (voxels, faces, edges and vertices), and allows capturing all the topological incidence and adjacency relations between cells by using only integer operations.We illustrate an application of this coordinate system on two tasks related with the repaired image: boundary reconstruction and computation of the Euler characteristic.

Tropical ideals

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing Giansiracusa and Giansiracusa [Equations of tropical varieties, Duke Math. J. 165 (2016), 3379–3433]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.

Regularity of harmonic maps from polyhedra to CAT(1) spaces

We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Holder regularity with Holder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the $$(n-2)$$ -skeleton, we improve the regularity to locally Lipschitz. Finally, for points $$x \in X^{(k)}$$ with $$k \le n-2$$ , we demonstrate that the Holder exponent depends on geometric and combinatorial data of the link of $$x \in X$$ .

On expansion and topological overlap

We give a detailed and easily accessible proof of Gromov’s Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map Xrightarrow mathbb {R}^d there exists a point pin mathbb {R}^d that is contained in the images of a positive fraction mu >0 of the d-cells of X. More generally, the conclusion holds if mathbb {R}^d is replaced by any d-dimensional piecewise-linear manifold M, with a constant mu that depends only on d and on the expansion properties of X, but not on M.

A duality in Buchsbaum rings and triangulated manifolds

Let $\Delta$ be a triangulated homology ball whose boundary complex is $\partial\Delta$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $\Delta$, $\mathbb F[\Delta]$, is isomorphic to the Stanley--Reisner module of the pair $(\Delta, \partial\Delta)$, $\mathbb F[\Delta,\partial \Delta]$. This result implies that an Artinian reduction of $\mathbb F[\Delta,\partial \Delta]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $\mathbb F[\Delta]$. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the $h"$-numbers of Buchsbaum complexes and use it to prove the monotonicity of $h"$-numbers for pairs of Buchsbaum complexes as well as the unimodality of $h"$-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold $g$-conjecture.

On the product of cocycles in a polyhedral complex

We construct an algorithm for multiplying cochains in a polyhedral complex. It depends on the choice of a linear functional on the ambient space. The cocycles form a subring in the ring of cochains, the coboundaries form an ideal in the ring of cocycles, and the quotient ring is the cohomology ring. The multiplication algorithm depends on the geometry of the cells of the complex. For simplicial complexes (the simplest geometry of cells), it reduces to the well-known Čech algorithm. Our algorithm is of geometric origin. For example, it applies in the calculation of mixed volumes of polyhedra and the construction of stable intersections of tropical varieties. In geometry it is customary to multiply cocycles with values in the exterior algebra of the ambient space. Therefore we assume that the ring of values is supercommutative.

Efficiently Storing Well-Composed Polyhedral Complexes Computed Over 3D Binary Images

A 3D binary image I can be naturally represented by a combinatorial-algebraic structure called cubical complex and denoted by Q(I), whose basic building blocks are vertices, edges, square faces and cubes. In Gonzalez-Diaz et al. (Discret Appl Math 183:59–77, 2015), we presented a method to “locally repair” Q(I) to obtain a polyhedral complex P(I) (whose basic building blocks are vertices, edges, specific polygons and polyhedra), homotopy equivalent to Q(I), satisfying that its boundary surface is a 2D manifold. P(I) is called a well-composed polyhedral complex over the picture I. Besides, we developed a new codification system for P(I), encoding geometric information of the cells of P(I) under the form of a 3D grayscale image, and the boundary face relations of the cells of P(I) under the form of a set of structuring elements. In this paper, we build upon (Gonzalez-Diaz et al. 2015) and prove that, to retrieve topological and geometric information of P(I), it is enough to store just one 3D point per polyhedron and hence neither grayscale image nor set of structuring elements are needed. From this “minimal” codification of P(I), we finally present a method to compute the 2-cells in the boundary surface of P(I).

A natural generalization of regular convex polyhedra

As a natural generalization of surfaces of Platonic solids, we define a class of polyhedra, called simple regular polyhedral BP-complexes, as a class of 2-dimensional polyhedral metric complexes satisfying certain conditions on their vertex sets, and we give a complete classification of such polyhedra. They are either the surface of a Platonic solid, a p-dodecahedron, a p-icosahedron, an m-covered regular n-gon for some m≧2 or a complete tripartite polygon.

On the product of cocycles in a polyhedral complex

Построен алгоритм умножения коцепей полиэдрального комплекса, зависящий от выбора линейного функционала на содержащем комплекс пространстве. Коциклы образуют подкольцо кольца коцепей, кограницы - идеал кольца коциклов, а фактор-кольцо является кольцом когомологий. Алгоритм умножения зависит от геометрии клеток комплекса. Если комплекс симплициален, т. е. геометрия клеток наиболее проста, то правило умножения коцепей сводится к известному алгоритму Чеха. Алгоритм имеет геометрическое происхождение. Например, он применяется для вычисления смешанных объемов многогранников и для построения стабильного пересечения тропических многообразий. В геометрии обычно перемножаются коциклы, принимающие значения во внешней алгебре пространства. Поэтому мы предполагаем, что кольцо значений суперкоммутативно. Библиография: 18 наименований.

Tropical Catalan subdivisions

We revisit the associahedral subdivision of the Pitman-Stanley polytope to provide geometric realizations of the ν-Tamari lattice of Préville-Ratelle and Viennot (which generalizes the m-Tamari lattice) as the dual of a triangulation of a polytope, as the dual of a mixed subdivision and as the edge-graph of a polyhedral complex induced by a tropical hyperplane arrangement. The method generalizes to type Bn.

The distortion dimension of $$\mathbb Q$$-rank 1 lattices

Let $$X=G/K$$ be a symmetric space of noncompact type and rank $$k\ge 2$$ . We prove that horospheres in X are Lipschitz $$(k-2)$$ -connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible $$\mathbb {Q}$$ -rank-1 lattice $$\Gamma $$ in a linear, semisimple Lie group G of $$\mathbb R$$ -rank k is $$k-1$$ . That is, given $$m< k-1$$ , a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) $$\Gamma $$ , and a $$(m+1)$$ -ball B in X (or G) filling S, there is a $$(m+1)$$ -ball $$B'$$ in $$\Gamma $$ filling S such that $${{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B$$ . In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension $$k-1$$ .