The excess degree of a polytope
- Authors: Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2018
- Type: Text , Journal article
- Relation: SIAM Journal on Discrete Mathematics Vol. 32, no. 3 (2018), p. 2011-2046, http://purl.org/au-research/grants/arc/DP180100602
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- Description: We define the excess degree \xi (P) of a d-polytope P as 2f1 - df0, where f0 and f1 denote the number of vertices and edges, respectively. This parameter measures how much P deviates from being simple. It turns out that the excess degree of a d-polytope does not take every natural number: the smallest possible values are 0 and d - 2, and the value d - 1 only occurs when d = 3 or 5. On the other hand, for fixed d, the number of values not taken by the excess degree is finite if d is odd, and the number of even values not taken by the excess degree is finite if d is even. The excess degree is then applied in three different settings. First, it is used to show that polytopes with small excess (i.e., \xi (P) < d) have a very particular structure: provided d ot = 5, either there is a unique nonsimple vertex, or every nonsimple vertex has degree d + 1. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Second, we characterize completely the decomposable d-polytopes with 2d + 1 vertices (up to combinatorial equivalence). Third, all pairs (f0, f1), for which there exists a 5-polytope with f0 vertices and f1 edges, are determined.
On the reconstruction of polytopes
- Authors: Doolittle, Joseph , Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2019
- Type: Text , Journal article
- Relation: Discrete and Computational Geometry Vol. 61, no. 2 (2019), p. 285-302. http://purl.org/au-research/grants/arc/DP180100602
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- Description: Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d- 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d> 3 there are two d-polytopes with d- 1 nonsimple vertices, isomorphic (d- 3) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
Almost simplicial polytopes : the lower and upper bound theorems
- Authors: Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2020
- Type: Text , Journal article
- Relation: Canadian Journal of Mathematics Vol. 72, no. 2 (2020), p. 537-556. http://purl.org/au-research/grants/arc/DP180100602
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- Description: We study -vertex -dimensional polytopes with at most one nonsimplex facet with, say, vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of, and, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where s = 0. We characterize the minimizers and provide examples of maximizers for any. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest. © 2018 Canadian Mathematical Society.
Chebyshev multivariate polynomial approximation and point reduction procedure
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2021
- Type: Text , Journal article
- Relation: Constructive Approximation Vol. 53, no. 3 (2021), p. 529-544
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Chebyshev multivariate polynomial approximation : alternance interpretation
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2018
- Type: Text , Book chapter
- Relation: 2016 Matrix Annals p. 177-182
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- Description: In this paper, we derive optimality conditions for Chebyshev approximation of multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions was developed in the late nineteenth and twentieth century. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). It is not clear, however, how to extend the notion of alternance to the case of multivariate functions. There have been several attempts to extend the theory of Chebyshev approximation to the case of multivariate functions. We propose an alternative approach, which is based on the notion of convexity and nonsmooth analysis.
Schur functions for approximation problems
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2020
- Type: Text , Book chapter
- Relation: 2018 Matrix Annals p. 331-337
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- Description: In this paper we propose a new approach to least squares approximation problems. This approach is based on partitioning and Schur function. The nature of this approach is combinatorial, while most existing approaches are based on algebra and algebraic geometry. This problem has several practical applications. One of them is curve clustering. We use this application to illustrate the results.
Minimum number of edges of polytopes with 2d + 2 vertices
- Authors: Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2022
- Type: Text , Journal article
- Relation: Electronic Journal of Combinatorics Vol. 29, no. 3 (2022), p.
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: We define two d-polytopes, both with 2d + 2 vertices and (d + 3)(d
Lower bound theorems for general polytopes
- Authors: Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2019
- Type: Text , Journal article
- Relation: European Journal of Combinatorics Vol. 79, no. (2019), p. 27-45
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: For a d-dimensional polytope with v vertices, d + 1 <= v <= 2d, we calculate precisely the minimum possible number of m-dimensional faces, when m = 1 or m >= 0.62d. This confirms a conjecture of Grunbaum, for these values of m. For v = 2d + 1, we solve the same problem when m = 1 or d - 2; the solution was already known for m = d - 1. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of m-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.
Polytopes close to being simple
- Authors: Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2020
- Type: Text , Journal article
- Relation: Discrete and Computational Geometry Vol. 64, no. 1 (2020), p. 200-215
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that d-polytopes with at most d- 2 nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2 and d- 2 , showing that certain polytopes with more than two nonsimple vertices are reconstructible from their graphs. In particular, we prove that reconstructibility from graphs also holds for d-polytopes with d+ k vertices and at most d- k+ 3 nonsimple vertices, provided k