Convex Geometry Research Papers - Academia.edu (original) (raw)

2007, Nature

Subducting slabs provide the main driving force for plate motion and flow in the Earth's mantle 1-4 , and geodynamic, seismic and geochemical studies offer insight into slab dynamics and subductioninduced flow 3-15 . Most previous geodynamic studies treat subduction zones as either infinite in trench-parallel extent 3,5,6 (that is, two-dimensional) or finite in width but fixed in space 7,16 . Subduction zones and their associated slabs are, however, limited in lateral extent (250-7,400 km) and their three-dimensional geometry evolves over time. Here we show that slab width controls two first-order features of plate tectonics-the curvature of subduction zones and their tendency to retreat backwards with time. Using three-dimensional numerical simulations of free subduction, we show that trench migration rate is inversely related to slab width and depends on proximity to a lateral slab edge. These results are young (#5 Myr), have a short slab (#150 km) and, together with collision zones, have been excluded from the trench migration calculations in . Trench migration rates for incipient subduction zones are presented in .

2008

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens the door to a number of extensions. The purpose of this paper is to initiate a systematic development of the theory, and to give a number of applications and examples.

1985, Geometriae Dedicata

2004, IEEE Transactions on Geoscience and Remote Sensing

Several of the more important endmember-finding algorithms for hyperspectral data are discussed and some of their shortcomings highlighted. A new algorithm-iterated constrained endmembers (ICE)-which attempts to address these shortcomings is introduced. An example of its use is given. There is also a discussion of the advantages and disadvantages of normalizing spectra before the application of ICE or other endmember-finding algorithms.

2010

The piece-wise convex multiple model endmember detection algorithm (P-COMMEND) and the Piece-wise Convex End-member detection (PCE) algorithm autonomously estimate many sets of endmembers to represent a hyperspectral image. A piece-wise convex model with several sets of endmembers is more effective for representing non-convex hyperspectral imagery over the standard convex geometry model (or linear mixing model). The terms of the objective function in P-COMMEND are based on geometric properties of the input data and the endmember estimates. In this paper, the P-COMMEND algorithm is extended to autonomously determine the number of sets of endmembers needed. The number of sets of endmembers, or convex regions, is determined by incorporating the competitive agglomeration algorithm into P-COMMEND. Results are shown comparing the Competitive Agglomeration P-COMMEND (CAP) algorithm to results found using the statistical PCE endmember detection method.

1986, SIAM Journal on Algebraic and Discrete …

Convexity in Graphs and Hypergraphs. [SIAM Journal on Algebraic and Discrete Methods 7, 433 (1986)]. Martin Farber, Robert E. Jamison. Abstract. We study several notions of abstract convexity in graphs and hypergraphs. In ...

arXiv preprint arXiv:0904.3350

Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that any semigroup in the lattice Z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results: we show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of Fujita approximation theorem for semigroups of integral points and graded algebras, which implies a generalization of this theorem for arbitrary linear series. Applications to intersection theory include a far-reaching generalization of the Kushnirenko theorem (from Newton polytope theory) and a new version of the Hodge inequality. We also give elementary proofs of the Alexandrov-Fenchel inequality in convex geometry and its analogue in algebraic geometry.

1995, Proceedings of the sixth …

It is well known that chordal graphs can be characterized via m-convexity. In this paper we introduce the notion of m 3 -convexity (a relaxation of m-convexity) which is closely related to semisimplicial orderings of graphs. We present new characterizations of HHD-free graphs via m 3 -convexity and obtain some results known from [B. Jamison and S. Olariu, Adv. Appl. Math., 9 (1988), pp. 364-376] as corollaries. Moreover, we characterize weak bipolarizable graphs as the graphs for which the family of all m 3 -convex sets is a convex geometry. As an application of our results we present a simple efficient criterion for deciding whether a HHD-free graph contains a r-dominating clique with respect to a given vertex radius function r.

2009, Journal of Algebraic Geometry

We give an algebraic construction of the positive intersection products of pseudo-effective classes first introduced in [BDPP], and use them to prove that the volume function on the Néron-Severi space of a projective variety is C 1 -differentiable, expressing its differential as a positive intersection product. We also relate the differential to the restricted volumes introduced in [ELMNP3, Ta]. We then apply our differentiability result to prove an algebro-geometric version of the Diskant inequality in convex geometry, allowing us to characterize the equality case of the Khovanskii-Teissier inequalities for nef and big classes.

2010, IEEE Transactions on Geoscience and Remote Sensing

A new hyperspectral endmember detection method that represents endmembers as distributions, autonomously partitions the input data set into several convex regions, and simultaneously determines endmember distributions and proportion values for each convex region is presented. Spectral unmixing methods that treat endmembers as distributions or hyperspectral images as piece-wise convex data sets have not been previously developed.

2000, Discrete Applied Mathematics

A game on a convex geometry is a real-valued function deÿned on the family L of the closed sets of a closure operator which satisÿes the ÿnite Minkowski-Krein-Milman property. If L is the boolean algebra 2 N then we obtain an n-person cooperative game. Faigle and Kern investigated games where L is the distributive lattice of the order ideals of the poset of players. We obtain two classes of axioms that give rise to a unique Shapley value for games on convex geometries. ?

2009, Linear Algebra and its Applications

2000, IEEE Transactions on Information Theory

The entropy per coordinate in a log-concave random vector of any dimension with given density at the mode is shown to have a range of just 1. Uniform distributions on convex bodies are at the lower end of this range, the distribution with i.i.d. exponentially distributed coordinates is at the upper end, and the normal is exactly in the middle. Thus in terms of the amount of randomness as measured by entropy per coordinate, any log-concave random vector of any dimension contains randomness that differs from that in the normal random variable with the same maximal density value by at most 1/2. As applications, we obtain an information-theoretic formulation of the famous hyperplane conjecture in convex geometry, entropy bounds for certain infinitely divisible distributions, and quantitative estimates for the behavior of the density at the mode on convolution. More generally, one may consider so-called convex or hyperbolic probability measures on Euclidean spaces; we give new constraints on entropy per coordinate for this class of measures, which generalize our results under the log-concavity assumption, expose the extremal role of multivariate Pareto-type distributions, and give some applications.

2009

Spectral mixture analysis is an important task for remotely sensed hyperspectral data interpretation. In spectral unmixing, both the determination of spectrally pure signatures (endmembers) and the unmixing process that interprets mixed pixels as combinations of endmembers are computationally expensive procedures. An exciting recent development in the field of commodity computing is the emergence of programmable graphics processing units (GPUs), which are now increasingly being used address the ever-growing computational requirements introduced by hyperspectral imaging applications. In this paper, we develop three new GPU-based implementations of endmember extraction algorithms: the pixel purity index (PPI), a kernel version of the PPI (KPPI), and the automatic morphological endmember extraction (AMEE) algorithm. We also provide a GPU-based implementation of the fully constrained linear spectral unmixing algorithm.

2006, Information Sciences

We propose a definition for the entropy of capacities defined on lattices. Classical capacities are monotone set functions and can be seen as a generalization of probability measures. Capacities on lattices address the general case where the family of subsets is not necessarily the Boolean lattice of all subsets. Our definition encompasses the classical definition of Shannon for probability measures, as well as the entropy of Marichal defined for classical capacities. Some properties and examples are given.

arXiv preprint arXiv:0804.4095

2004, Remote Sensing of Environment

This paper reports on the use of linear spectral mixture analysis for the retrieval of canopy leaf area index (LAI) in three flux tower sites in the Boreal Ecosystem-Atmosphere Study (BOREAS) southern study area: Old Black Spruce, Old Jack Pine, and Young Jack Pine (SOBS, SOJP, and SYJP). The data used were obtained by the Compact Airborne Spectrographic Imager (CASI) with a spatial resolution of 2 m in the winter of 1994. The convex geometry method was used to select the endmembers: sunlit crown, sunlit snow, and shadow. Along transects for these flux tower sites, the fraction of sunlit snow was found to have a higher correlation with the field-measured canopy LAI than the fraction of sunlit crown or the fraction of shadow. An empirical equation was obtained to describe the relation between canopy LAI and the fraction of sunlit snow. There is a strong correlation between the estimated LAI and the field-measured LAI along transects (with R 2 values of 0.54, 0.71, and 0.60 obtained for the SOBS, SYJP, and SOJP sites, respectively). The estimated LAI for the whole tower site is consistent with that obtained by the inversion of a canopy model in our previous study where values of 0.94, 0.92, and 0.63 were obtained for R 2 for the SOBS, SYJP and SOJP sites, respectively.

2011, Comptes Rendus Mathematique

We develop an information-theoretic perspective on some questions in convex geometry, providing for instance a new equipartition property for log-concave probability measures, some Gaussian comparison results for log-concave measures, an entropic formulation of the hyperplane conjecture, and a new reverse entropy power inequality for log-concave measures analogous to V. Milman's reverse Brunn-Minkowski inequality. * S. G. Bobkov is with the

2011, Journal of Mathematical Imaging and Vision

We propose a novel approach to reconstruct shapes from digital data. Contrarily to most methods, reconstructed shapes are smooth with a well-defined curvature field and have the same digitization as the input data: the range of application we have in mind is especially postprocessing to image segmentation where labelled regions are digital objects. For this purpose, we introduce three new algorithms to regularize digital contours based on the minimization of Willmore energy: our first algorithm is based on tools coming from discrete geometry, the second is related to convex geometry while the third approach is a constrained phase field minimization. The three algorithms are described in details and the convergence of the phase field approach is investigated. We present a comparative evaluation of all three methods, in terms of the accuracy of curvature estimators and computation time.

2007, International Journal of Approximate Reasoning

We use standard results from convex geometry to obtain representations of the prior and posterior degrees of imprecision in terms of width functions and difference bodies. These representations are used to construct algorithms for the... more

1988, Order

We develop a representation theory for convex geometries and meet distributive lattices in the spirit of Birkhoff's theorem characterizing distributive lattices. The results imply that every convex geometry on a set X has a canonical representation as a poset labelled by elements of X. These resuits are related to recent work of Korte and Lovasz on antimatroids. We also compute the convex dimension of a convex geometry.

2008, European Journal of Combinatorics

We use the Steiner distance to define a convexity in the vertex set of a graph, which has a nice behavior in the well-known class of HHD-free graphs. For this graph class, we prove that any Steiner tree of a vertex set is included into the geodesical convex hull of the set, which extends the well-known fact that the Euclidean convex hull contains at least one Steiner tree for any planar point set. We also characterize the graph class where Steiner convexity becomes a convex geometry, and provide a vertex set that allows us to rebuild any convex set, using convex hull operation, in any graph. J. Cáceres), almar@ual.es (A. Márquez), mpuertas@ual.es (M.L. Puertas). 1 Tel.: +34 950 015 665; fax: +34 950 015 147. 0195-6698/$ -see front matter c

2008, Computers & Graphics

Collision detection (CD) between two objects is a complex task which demands spatial decomposition or bounding volume hierarchy in order to classify the features of the objects for a successful intersection test. Moreover, when the objects are complex and have concavities, holes, etc., the complexity of the decomposition increases and it is necessary to find a method which detects all special cases and degenerations.

A real semisimple Lie algebra g admits a Cartan involution, θ , for which the corresponding eigenspace decomposition g=k+p has the property that all operators ad X , X∈p are diagonalizable over R . We call such elements hyperbolic, and the elements X∈k are elliptic in the sense that ad X is semisimple with purely imaginary eigenvalues. The pairs (g,θ) are examples of symmetric Lie algebras, i.e., Lie algebras endowed with an involutive automorphism, such that the -1 -eigenspace of θ contains only hyperbolic elements. Let (g,τ ) be a symmetric Lie algebra and g= h+q the corresponding eigenspace decomposition for τ . The existence of "enough" hyperbolic elements in q is important for the structural analysis of symmetric Lie algebras in terms of root decompositions with respect to abelian subspaces of q consisting of hyperbolic elements. We study the convexity properties of the action of Inn g (h) on the space q . The key role will be played by those invariant convex subsets of q whose interior points are hyperbolic.

2006, Journal of Functional Analysis

Let W be a finite Coxeter group acting linearly on R n . In this article we study support properties of W -invariant partial differential operator D on R n with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W . In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W . We show that conv(supp Df ) = conv(supp f ) holds for all compactly supported smooth functions f so that conv(supp f ) is W -invariant. The main tools in the proof are Holmgren's uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented.

2008, Journal of Mathematical Sciences

Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold Cartesian product of the max-plus semiring: It is known that one can separate a vector from a closed subsemimodule that does not contain it. Here we establish a more general separation theorem, which applies to any finite collection of closed subsemimodules with a trivial intersection. The proof of this theorem involves specific nonlinear operators, called here cyclic projectors on idempotent semimodules. These are analogues of the cyclic nearest-point projections known in convex analysis. We obtain a theorem that characterizes the spectrum of cyclic projectors on idempotent semimodules in terms of a suitable extension of Hilbert's projective metric. We also deduce as a corollary of our main results the idempotent analogue of Helly's theorem.

2009, Contemporary Mathematics

Max cones are the subsets of the nonnegative orthant R n + of the n-dimensional real space R n closed under scalar multiplication and componentwise maximisation. Their study is motivated by some practical applications which arise in discrete event systems, optimal scheduling and modelling of synchronization problems in multiprocessor interactive systems. We investigate the geometry of max cones, concerning the role of the multiorder principle, the Kleene stars, and the cyclic projectors. The multiorder principle is closely related to the set covering conditions in max algebra, and gives rise to important analogues of some theorems of convex geometry. We show that, in particular, this principle leads to a convenient representation of certain nonlinear projectors onto max cones. The Kleene stars are fundamental in max algebra since they accumulate weights of optimal paths and yield generators for max-algebraic eigenspaces of matrices. We examine the role of their column spans called Kleene cones, as building blocks in the Develin-Sturmfels cellular decomposition. Further we show that the cellular decomposition gives rise to new max-algebraic objects which we call row and column Kleene stars. We relate these objects to the maxalgebraic pseudoinverses of matrices and to tropical versions of the colourful Carathéodory theorem. The cyclic projectors are specific nonlinear operators which lead to the so-called alternating method for finding a solution to homogeneous two-sided systems of max-linear equations. We generalize the alternating method to the case of homogeneous multi-sided systems, and we give a proof, which uses the cellular decomposition idea, that the alternating method converges in a finite number of iterations to a positive solution of a multi-sided system if a positive solution exists. We also present new bounds on the number of iterations of the alternating method, expressed in terms of the Hilbert projective distance between max cones.

1998, Mathematical Social Sciences

In this paper, we introduce the Banzhaf power indices for simple games on convex geometries. We define the concept of swing for these structures, obtaining convex swings. The number of convex swings and the number of coalitions such that a player is an extreme point are the basic tools to define the convex Banzhaf indices, one normalized and other probabilistic. We obtain a family of axioms that give rise to the Banzhaf indices. In the last section, we present a method to calculate the convex Banzhaf indices with the computer program Mathematica, and we apply this to compute power indices in the Spanish and Catalan parliaments and in the Council of Ministers of the European Union. 

2011, European Journal of Operational Research

The notion of interaction among a set of players has been defined on the Boolean lattice and Cartesian products of lattices. The aim of this paper is to extend this concept to combinatorial structures with forbidden coalitions. The set of feasible coalitions is supposed to fulfil some general conditions. This general representation encompasses convex geometries, antimatroids, augmenting systems and distributive lattices. Two axiomatic characterizations are obtained. They both assume that the Shapley value is already defined on the combinatorial structures. The first one is restricted to pairs of players and is based on a generalization of a recursivity axiom that uniquely specifies the interaction index from the Shapley value when all coalitions are permitted. This unique correspondence cannot be maintained when some coalitions are forbidden. From this, a weak recursivity axiom is defined. We show that this axiom together with linearity and dummy player are sufficient to specify the interaction index. The second axiomatic characterization is obtained from the linearity, dummy player and partnership axioms. An interpretation of the interaction index in the context of surplus sharing is also proposed. Finally, our interaction index is instantiated to the case of games under precedence constraints.

2012, Medical Engineering & Physics

Failure of ultra-high molecular weight polyethylene components after total disc replacements in the lumbar spine has been reported in several retrieval studies, but immediate biomechanical evidence for those mechanical failures remained unclear. Current study aimed to investigate the failure mechanisms of commercial lumbar disc prostheses and to enhance the biomechanical performances of polyethylene components by modifying the articulating surface into a convex geometry. Modified compressive-shearing tests were utilized in finite element analyses for comparing the contact, tensile, and shearing stresses on two commercial disc prostheses and on a concave polyethylene design. The influence of radial clearance on stress distributions and prosthetic stability were considered. The modified compressive-shearing test revealed the possible mechanisms for transverse and radial cracks of polyethylene components, and would be helpful in observing the mechanical risks in the early design stage. Additionally, the concave polyethylene component exhibited lower contact and shearing stresses and more acceptable implant stability when compared with the convex polyethylene design through all radial clearances. Use of a concave polyethylene component in lumbar disc replacements decreased the risk of transverse and radial cracks, and also helped to maintain adequate stability. This design concept should be considered in lumbar disc implant designs in the future.

2007

We provide a new characterization of convex geometries via a multivariate version of an identity that was originally proved, in a special case arising from the k-SAT problem, by Maneva, Mossel and Wainwright. We thus highlight the connection between various characterizations of convex geometries and a family of removal processes studied in the literature on random structures.

2008, arXiv preprint arXiv:0812.4688

The well-known Bernstein-Kušhnirenko theorem from the theory of Newton polyhedra relates algebraic geometry and the theory of mixed volumes. Recently the authors have found a far-reaching generalization of this theorem to generic systems of algebraic equations on any quasi-projective variety. In the present note we review these results and their applications to algebraic geometry and convex geometry.

2009, Czechoslovak Mathematical Journal

In this paper we first study what changes occur in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then, we study the set of all convex geometries which have the same poset of joinirreducible elements. We show that this set-ordered by set inclusion-is a ranked join-semilattice and we characterize its cover relation. We prove that the lattice of all ideals of a given poset P is the only convex geometry having a poset of join-irreducible elements isomorphic to P if and only if the width of P is less than 3. Finally, we give an algorithm for computing all convex geometries having the same poset of join-irreducible elements.

2008, Proceedings of Symposia in Pure Mathematics

Tube formulas (by which we mean an explicit formula for the volume of an ε-neighbourhood of a subset of a suitable metric space) have been used in many situations to study properties of the subset. For smooth submanifolds of Euclidean space, this includes Weyl's celebrated results on spectral asymptotics, and the subsequent relation between curvature and spectrum. Additionally, a tube formula contains information about the dimension and measurability of rough sets. In convex geometry, the tube formula of a convex subset of Euclidean space allows for the definition of certain curvature measures. These measures describe the curvature of sets which are not too irregular to support derivatives. In this survey paper, we describe some recent advances in the development of tube formulas for self-similar fractals, and their applications and connections to the other topics mentioned here.

2007, Journal of Applied and Industrial Mathematics

This is an overview of merging the techniques of vector lattice theory and convex geometry.

2000, Discrete & Computational Geometry

We prove a formula conjectured by Ahrens, Gordon, and McMahon for the number of interior points for a point con guration in R d . Our method is to show that the formula can be interpreted as a sum of Euler characteristics of certain complexes associated with the point con guration, and then compute the homology of these complexes. This method extends to other examples of convex geometries. We sketch these applications, replicating an earlier result of Gordon, and proving a new result related to ordered sets.

2004

In this paper we study two lattices of significant particular closure systems on a finite set, namely the union stable closure systems and the convex geometries. Using the notion of (admissible) quasi-closed set and of (deletable) closed set we determine the covering relation ≺ of these lattices and the changes induced, for instance, on the irreducible elements when one goes from C to C where C and C are two such closure systems satisfying C ≺ C. We also do a systematic study of these lattices of closure systems, characterizing for instance their join-irreducible and their meet-irreducible elements.

2010, Physica B: Condensed Matter

In this work we present advances in the design of lenses based on left-handed extraordinary transmission metamaterials which provide high transmission and focusing despite the subwavelength size of their constituent apertures. Due to the effective negative index of refraction of the close-stack of subwavelength hole arrays, concave profiles are required for focusing instead of the convex geometries of dielectric lenses. An analysis of the foci produced by plano-and bi-concave lens is carried out.

2014, European Journal of Combinatorics

A closure system with the anti-exchange axiom is called a convex geometry. One geometry is called a sub-geometry of the other if its closed sets form a sublattice in the lattice of closed sets of the other. We prove that convex geometries of relatively convex sets in n-dimensional vector space and their finite sub-geometries satisfy the n-Carousel Rule, which is the strengthening of the n-Carathéodory property. We also find another property, that is similar to the simplex partition property and does not follow from 2-Carusel Rule, which holds in sub-geometries of 2-dimensional geometries of relatively convex sets.

2010, Discrete Mathematics

A function F defined on all subsets of a finite ground set E is quasi-

2007, Advances in Mathematics

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K ⊂ R d , which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by 2 d smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. More precisely, we show that for every centrally symmetric convex body K ⊂ R d one has

2009, Order

The paper puts forth a theory of choice functions in a neat way connecting it to a theory of extensive operators and neighborhood systems. We consider four classes of heritage choice functions satisfying the conditions M, N, W, and C.

2008

We introduce and study a new class of eps\epseps-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how eps\epseps-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, and find applications to geometric tomography. We introduce the notion of a circular projection

2010

An endmember detection and spectral unmixing algorithm that uses both spatial and spectral information is presented. This method, Spatial Piece-wise Convex Multiple Model Endmember Detection (Spatial P-COMMEND), autonomously estimates multiple sets of endmembers and performs spectral unmixing for input hyperspectral data. Spatial P-COMMEND does not restrict the estimated endmembers to define a single convex region during spectral unmixing. Instead, a piece-wise convex representation is used that can effectively represent non-convex hyperspectral data. Spatial P-COMMEND drives neighboring pixels to be unmixed by the same set of endmembers encouraging spatially-smooth unmixing results.

2004, … of the 10th International Conference on …

In this paper we extend our geometric approach to the theory of evidence
in order to include other important classes of nite fuzzy measures.
In particular we describe the geometric counterparts of possibility measures or fuzzy sets, represented as consonant belief functions.
The correspondence between chains of subsets and convex sets of consonant functions is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions
in terms of the notion of simplicial complex.

1985, Communications in Mathematical Physics

The main result is a representation theorem which shows that, for a large class of quantum logics, a quantum logic, β, is isomorphic to the lattice of projective faces in a suitable convex set K. As an application we extend our earlier results , which, subject to countability conditions, gave a geometric characterization of those quantum logics which are isomorphic to the projection lattice of a von Neumann algebra or a JBW-algebva.

1993, Algebra Universalis

In this paper we show that the set of closure relations on a nite poset P forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice is dually isomorphic to the lattice of closed sets in a convex geometry (in the sense of Edelman and Jamison EJ]). We also characterize the modular elements of this lattice (when P has a greatest element) and compute its characteristic polynomial.

2007, Discrete Mathematics

We consider matroidal structures on convex geometries, which we call cg-matroids. The concept of a cg-matroid is closely related to but different from that of a supermatroid introduced by Dunstan, Ingleton, and Welsh in 1972. Distributive supermatroids or poset matroids are supermatroids defined on distributive lattices or sets of order ideals of posets. The class of cg-matroids includes distributive supermatroids (or poset matroids). We also introduce the concept of a strict cg-matroid, which turns out to be exactly a cg-matroid that is also a supermatroid. We show characterizations of cg-matroids and strict cg-matroids by means of the exchange property for bases and the augmentation property for independent sets. We also examine submodularity structures of strict cg-matroids.

2008

Let G be a geometric graph in the plane whose edges may be curves. For two arbitrary points on its edges, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set S, we would like to know the smallest possible dilation of any graph that contains the given points on its edges. We call this infimum the dilation of S and denote it by δ(S).

Mathematical Notes

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