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70101 Pure Mathematics
4Polytopes
30802 Computation Theory and Mathematics
201 Mathematical Sciences
2Banach space
10103 Numerical and Computational Mathematics
113 Education
12005 Literary Studies
13 space problem
152B05 (Secondary)
152B10
152B11 (Primary)
1Algorithms
1Banach-spaces
1Calculus
1Classification
1Complemented subspace
1Convex-sets
1Decomposable
1Excess degree

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Reducible polytopes

**Authors:**Yost, David**Date:**2005**Type:**Text , Conference paper**Relation:**Paper presented at the Sixteenth Australasian Workshop on Combinatorial Algorithms, 18-21 September 2005, Ballarat, Australia, Ballarat, Victoria : 18th - 21st September, 2005**Full Text:****Reviewed:****Description:**E1**Description:**2003001436

**Authors:**Yost, David**Date:**2005**Type:**Text , Conference paper**Relation:**Paper presented at the Sixteenth Australasian Workshop on Combinatorial Algorithms, 18-21 September 2005, Ballarat, Australia, Ballarat, Victoria : 18th - 21st September, 2005**Full Text:****Reviewed:****Description:**E1**Description:**2003001436

A Different Johnson-Lindenstrauss Space

**Authors:**Yost, David**Date:**2008**Type:**Text , Journal article**Relation:**New Zealand Journal of Mathematics Vol. 37, no. (2008), p. 47-49**Full Text:**false**Reviewed:****Description:**We exhibit a non-trivial twisted sum of c0 with a Hilbert space, which is not isomorphic to the example constructed by Johnson and Lindenstrauss in 1974. In fact, no non-separable subspace of either example is isomorphic to any subspace of the other.**Description:**2003006488

Uniformly non-hexagonal Banach spaces

**Authors:**Yost, David**Date:**2007**Type:**Text , Conference paper**Relation:**Paper presented at 8th International Conference on Fixed Point Theory and its Applications, ICFPTA2007, Chiang Mai, Thailand : 16th-22nd July 2007**Full Text:**false**Description:**2003006614

Some indecomposable polyhedra

**Authors:**Yost, David**Date:**2007**Type:**Text , Journal article**Relation:**Optimization Vol. 56, no. 5-6 (2007), p. 715-724**Full Text:****Reviewed:****Description:**We complete the classification, in terms of decomposability, of all combinatorial types of polytopes with 14 or fewer edges. Recall that a polytope P is said to be decomposable if it is equal to a Minkowski sum [image omitted] of two polytopes Q and R which are not similar to P. Our main contribution here is to consider the 42 types of polyhedra with 8 faces and 8 vertices. It turns out that 34 of these are always indecomposable, and 5 are always decomposable. The remaining 3 are ambiguous, i.e. each of them has both decomposable and indecomposable geometric realizations.**Description:**C1**Description:**2003004904

**Authors:**Yost, David**Date:**2007**Type:**Text , Journal article**Relation:**Optimization Vol. 56, no. 5-6 (2007), p. 715-724**Full Text:****Reviewed:****Description:**We complete the classification, in terms of decomposability, of all combinatorial types of polytopes with 14 or fewer edges. Recall that a polytope P is said to be decomposable if it is equal to a Minkowski sum [image omitted] of two polytopes Q and R which are not similar to P. Our main contribution here is to consider the 42 types of polyhedra with 8 faces and 8 vertices. It turns out that 34 of these are always indecomposable, and 5 are always decomposable. The remaining 3 are ambiguous, i.e. each of them has both decomposable and indecomposable geometric realizations.**Description:**C1**Description:**2003004904

Extending of L-infinity-valued operators under a twisted light

- Cabello Sanchez, Felix, Castillo, Jesus, Moreno, Y, Yost, David

**Authors:**Cabello Sanchez, Felix , Castillo, Jesus , Moreno, Y , Yost, David**Date:**2002**Type:**Text , Conference paper**Relation:**Paper presented at Functional analysis and its applications. Proceedings of the International Conference. dedicated to the 110th anniversary of Stefan Banach, Lviv, Ukraine : 28th - 31st May, 2002 p. 59-70**Full Text:****Reviewed:****Description:**E1

**Authors:**Cabello Sanchez, Felix , Castillo, Jesus , Moreno, Y , Yost, David**Date:**2002**Type:**Text , Conference paper**Relation:**Paper presented at Functional analysis and its applications. Proceedings of the International Conference. dedicated to the 110th anniversary of Stefan Banach, Lviv, Ukraine : 28th - 31st May, 2002 p. 59-70**Full Text:****Reviewed:****Description:**E1

Integration: Reversing traditional pedagogy

**Authors:**Yost, David**Date:**2008**Type:**Text , Journal article**Relation:**Australian Senior Mathematics Journal Vol. 22, no. 2 (2008), p. 37-40**Full Text:**false**Reviewed:****Description:**The author recommends teaching integration first, and move on to differential calculus only after a sound understanding of integration has been attained. A rationale for teaching integrals before derivatives in a first calculus unit intended primarily for engineering students at the University of Ballarat is presented. It is concluded that students fail to see the connection between getting the right answer and understanding what they are doing. It is noted that the formal manipulations required for high school differential calculus questions are simpler than those for integral calculus.

Lipschitz selections for multifunctions

**Authors:**Yost, David**Date:**2010**Type:**Text , Conference proceedings**Full Text:**false

The excess degree of a polytope

- Pineda-Villavicencio, Guillermo, Ugon, Julien, Yost, David

**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**Full Text:****Reviewed:****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 \not = 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.

**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**Full Text:****Reviewed:****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 \not = 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.

Colocality and twisted sums of Banach spaces

- Jebreen, H. M., Jamjoom, F. B. H., Yost, David

**Authors:**Jebreen, H. M. , Jamjoom, F. B. H. , Yost, David**Date:**2006**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 323, no. 2 (2006), p. 864-875**Full Text:****Reviewed:****Description:**Using the relation between subspaces of Banach spaces and quotients of their duals, we introduce the concept of colocality to give a new method that guarantees the existence of nontrivial twisted sums in which finite quotients play a major role (Theorem 1.7). An interesting point is that no restrictions are imposed on the quotients, only on the various subspaces. New examples of nontrivial twisted sums are given.**Description:**C1**Description:**2003001831

**Authors:**Jebreen, H. M. , Jamjoom, F. B. H. , Yost, David**Date:**2006**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 323, no. 2 (2006), p. 864-875**Full Text:****Reviewed:****Description:**Using the relation between subspaces of Banach spaces and quotients of their duals, we introduce the concept of colocality to give a new method that guarantees the existence of nontrivial twisted sums in which finite quotients play a major role (Theorem 1.7). An interesting point is that no restrictions are imposed on the quotients, only on the various subspaces. New examples of nontrivial twisted sums are given.**Description:**C1**Description:**2003001831

Decomposability of polytopes

- Przeslawski, Krzysztof, Yost, David

**Authors:**Przeslawski, Krzysztof , Yost, David**Date:**2008**Type:**Text , Journal article**Relation:**Discrete & Computational Geometry Vol. 39, no. 1-3 (Mar 2008), p. 460-468**Full Text:****Reviewed:****Description:**A known characterization of the decomposability of polytopes is reformulated in a way which may be more computationally convenient, and a more transparent proof is given. New sufficient conditions for indecomposability are then deduced, and illustrated with some examples.**Description:**C1

**Authors:**Przeslawski, Krzysztof , Yost, David**Date:**2008**Type:**Text , Journal article**Relation:**Discrete & Computational Geometry Vol. 39, no. 1-3 (Mar 2008), p. 460-468**Full Text:****Reviewed:****Description:**A known characterization of the decomposability of polytopes is reformulated in a way which may be more computationally convenient, and a more transparent proof is given. New sufficient conditions for indecomposability are then deduced, and illustrated with some examples.**Description:**C1

Twisted sums with C(K) spaces

- Cabello Sanchez, Felix, Castillo, Jesus, Kalton, Nigel, Yost, David

**Authors:**Cabello Sanchez, Felix , Castillo, Jesus , Kalton, Nigel , Yost, David**Date:**2003**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 355, no. (2003), p. 4523-4541**Full Text:****Reviewed:****Description:**If X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 -->**Description:**C1**Description:**2003002201

**Authors:**Cabello Sanchez, Felix , Castillo, Jesus , Kalton, Nigel , Yost, David**Date:**2003**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 355, no. (2003), p. 4523-4541**Full Text:****Reviewed:****Description:**If X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 -->**Description:**C1**Description:**2003002201

Quasilinear Mappings, M-Ideals and Popyhedra

**Authors:**Yost, David**Date:**2012**Type:**Text , Conference paper**Relation:**Operators and Matrices Vol. 6, p. 279-286**Full Text:****Reviewed:****Description:**We survey the connection between two results from rather different areas: failure of the 3-space property for local convexity (and other properties) within the category of quasi-Banach spaces, and the irreducibility (in the sense of Minkowski difference) of large families of finite dimensional polytopes.**Description:**C1

**Authors:**Yost, David**Date:**2012**Type:**Text , Conference paper**Relation:**Operators and Matrices Vol. 6, p. 279-286**Full Text:****Reviewed:****Description:**We survey the connection between two results from rather different areas: failure of the 3-space property for local convexity (and other properties) within the category of quasi-Banach spaces, and the irreducibility (in the sense of Minkowski difference) of large families of finite dimensional polytopes.**Description:**C1

On the reconstruction of polytopes

- Doolittle, Joseph, Nevo, Eran, Pineda-Villavicencio, Guillermo, Ugon, Julien, Yost, David

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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.

More indecomposable polyhedra

- Przesławski, Krzysztof, Yost, David

**Authors:**Przesławski, Krzysztof , Yost, David**Date:**2016**Type:**Text , Journal article**Relation:**Extracta Mathematicae Vol. 31, no. 2 (2016), p. 169-188**Full Text:****Reviewed:****Description:**We apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of Minkowski decomposability. Various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a significant class of polytopes. We illustrate further the power of these techniques, compared with the traditional method of examining triangular faces, with several applications. In any dimension d 6= 2, we show that of all the polytopes with d2 + 1 2 d or fewer edges, only one is decomposable. In 3 dimensions, we complete the classification, in terms of decomposability, of the 260 combinatorial types of polyhedra with 15 or fewer edges.**Description:**We apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of Minkowski decomposability. Various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a significant class of polytopes. We illustrate further the power of these techniques, compared with the traditional method of examining triangular faces, with several applications. In any dimension $d\neq 2$, we show that of all the polytopes with $d^2+\frac{d}{2}$ or fewer edges, only one is decomposable. In 3 dimensions, we complete the classification, in terms of decomposability, of the 260 combinatorial types of polyhedra with 15 or fewer edges.

**Authors:**Przesławski, Krzysztof , Yost, David**Date:**2016**Type:**Text , Journal article**Relation:**Extracta Mathematicae Vol. 31, no. 2 (2016), p. 169-188**Full Text:****Reviewed:****Description:**We apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of Minkowski decomposability. Various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a significant class of polytopes. We illustrate further the power of these techniques, compared with the traditional method of examining triangular faces, with several applications. In any dimension d 6= 2, we show that of all the polytopes with d2 + 1 2 d or fewer edges, only one is decomposable. In 3 dimensions, we complete the classification, in terms of decomposability, of the 260 combinatorial types of polyhedra with 15 or fewer edges.**Description:**We apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of Minkowski decomposability. Various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a significant class of polytopes. We illustrate further the power of these techniques, compared with the traditional method of examining triangular faces, with several applications. In any dimension $d\neq 2$, we show that of all the polytopes with $d^2+\frac{d}{2}$ or fewer edges, only one is decomposable. In 3 dimensions, we complete the classification, in terms of decomposability, of the 260 combinatorial types of polyhedra with 15 or fewer edges.

Lower bound theorems for general polytopes

- Pineda-Villavicencio, Guillermo, Ugon, Julien, Yost, David

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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.

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