Polytopes in Measurement and Data Analysis. Review ofLectures on Polytopes,by Günter M. Ziegler (original) (raw)
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Polytopes in Measurement and Data Analysis. Review of Lectures on Polytopes,by G�nter M. Ziegler
J Math Psychol, 1997
Reviewed by Reinhard Suck Gu nter Ziegler is Professor of Mathematics at the Technische Universita t Berlin. Previously he held positions at Augsburg University and at ZIB Berlin. He received his Ph.D. in Mathematics from MIT in 1987. He was awarded the Gerhard Hess-Prize of the German Science Foundation (DFG) in 1994. His primary research interests are topological combinatorics, discrete geometry, and linear and combinatorial optimization. He is a coauthor (with A. Bjo rner, M. Las Vergnas, B. Sturmfels, and N. White) of the book Oriented Matroids (Cambridge University Press, 1993). Reinhard Suck of Akademischer Rat at the University of Osnabru ck. His primary research interests are measurement and probabilistic scaling with a bias towards discrete mathematics. He is coeditor with Eddy Roskam of Progress in Mathematical Psychology (Amsterdam: North Holland, 1987).
A Glimpse into Continuous Combinatorics of Posets, Polytopes, and Matroids
Journal of Mathematical Sciences, 2020
Following [Ži98] we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples reviewed in this paper are an Euler formula for a class of 'continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of 'continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of the 'homotopy complementation formula' for topological posets and its relation to the results of Kallel and Karoui about 'weighted barycenter spaces' and a conjecture of Vassiliev about simplicial resolutions of singularities. We also include an extension of the index inequality (Sarkaria's inequality) based on interpreting diagrams of spaces as continuous posets.
Combinatorial Structure of the Polytope of 2-Additive Measures
IEEE Transactions on Fuzzy Systems, 2019
In this paper we study the polytope of 2-additive measures, an important subpolytope of the polytope of fuzzy measures. For this polytope, we obtain its combinatorial structure, namely the adjacency structure and the structure of 2-dimensional faces, 3-dimensional faces, and so on. Basing on this information, we build a triangulation of this polytope satisfying that all simplices in the triangulation have the same volume. As a consequence, this allows a very simple and appealing way to generate points in a random way in this polytope, an interesting problem arising in the practical identification of 2-additive measures. Finally, we also derive the volume, the centroid, and some properties concerning the adjacency graph of this polytope.
Polytopes: Abstract, Convex and Computational
Polytopes: Abstract, Convex and Computational, 1994
Convex and Computational NATO ASI Series Advanced Science Institutes Series A Series presenting the results of activities sponsored by the NATO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities.
Classification of Finite Metric Spaces and Combinatorics of Convex Polytopes
Arnold Mathematical Journal, 2015
We describe the canonical correspondence between finite metric spaces and symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those polytopes. Keywords Finite metric space • Convex polytope • Kantorovich-Rubinstein norm • Classification 1 Introductory Remark We will discuss problems with very elementary formulations that concern the most popular notions in mathematics: metric spaces, convex geometry, combinatorics of polytopes and Kantorovich's optimal transportation. According to Arnold's classification, there are two ways to introduce a new subject: the first way (he called it the "Russian tradition") is to start with "the simplest non-trivial partial case"-I will use this approach. The second and the opposite tradition, which I also like very much (he called it "Bourbaki's tradition") is to start with an "extremely general case that is impossible to further generalize". So my metric spaces will be finite, polytopes will be finite-dimensional etc. but all the notions and problems have infinite, infinite-dimensional, and continuous analogs. To the memory of Dima Arnold.
8. Cardinality Homogeneous Set Systems, Cycles in Matroids, and Associated Polytopes
The Impact of Manfred Padberg and His Work, 2004
A subset C of the power set of a finite set E is called cardinality homogeneous if, whenever C contains some set F , C contains all subsets of E of cardinality |F |. Examples of such set systems C are the sets of all even or of all odd cardinality subsets of E, or, for each uniform matroid, its set of circuits and its set of cycles. With each cardinality homogeneous set system C, we associate the polytope P (C), the convex hull of the incidence vectors of all sets in C. We provide a complete and nonredundant linear description of P (C). We show that a greedy algorithm optimizes any linear function over P (C), construct, by a dual greedy procedure, an explicit optimum solution of the dual linear program, and describe a polynomial time separation algorithm for the class of polytopes of type P (C).
On the polytope of non-additive measures
Fuzzy Sets and Systems, 2008
In this paper we deal with the problem of studying the structure of the polytope of non-additive measures for finite referential sets. We give a necessary and sufficient condition for two extreme points of this polytope to be adjacent. We also show that it is possible to find out in polynomial time whether two vertices are adjacent. These results can be extended to the polytope given by the convex hull of monotone Boolean functions. We also give some results about the facets and edges of the polytope of non-additive measures; we prove that the diameter of the polytope is 3 for referentials of three elements or more. Finally, we show that the polytope is combinatorial and study the corresponding properties; more concretely, we show that the graph of non-additive measures is Hamilton connected if the cardinality of the referential set is not 2.
A Polytope Related to Empirical Distributions, Plane Trees, Parking Functions, and the Associahedron
Discrete & Computational Geometry, 2002
The volume of the n-dimensional polytope Π n (x) := {y ∈ R n : y i ≥ 0 and y 1 + · · · + y i ≤ x 1 + · · · + x i for all 1 ≤ i ≤ n} for arbitrary x := (x 1 , . . . , x n ) with x i > 0 for all i defines a polynomial in variables x i which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two different polytopal subdivisions of Π n (x). The first of these subdivisions generalizes to a class of polytopes called sections of order cones. In the second subdivision, the chambers are indexed in a natural way by rooted binary trees with n + 1 vertices, and the configuration of these chambers provides a representation of another polytope with many applications, the associahedron.
Israel Journal of Mathematics, 1982
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Uniform decompositions of polytopes
Applicationes Mathematicae, 2006
We design a method of decomposing convex polytopes into simpler polytopes. This decomposition yields a way of calculating exactly the volume of the polytope, or, more generally, multiple integrals over the polytope, which is equivalent to the way suggested in [9], based on Fourier-Motzkin elimination ([10, pp. 155-157]). Our method is applicable for finding uniform decompositions of certain natural families of polytopes. Moreover, this allows us to find algorithmically an analytic expression for the distribution function of a random variable of the form d i=1 c i X i , where (X 1 ,. .. , X d) is a random vector, uniformly distributed in a polytope.