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The Cayley Trick, Lifting Subdivisions And The BohneDress Theorem On Zonotopal Tilings
 J. EUR. MATH. SOC
, 1999
"... In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an orderpreserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum A 1 + \Delta \Delta \Delta +A r of point configurations and of coherent polyhedral subdivisions o ..."
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Cited by 31 (13 self)
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In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an orderpreserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum A 1 + \Delta \Delta \Delta +A r of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding C (A 1 ; : : : ; A r ). In this paper we extend this correspondence in a natural way to cover also noncoherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the BohneDress Theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos.
The generalized Baues problem
, 1998
"... Abstract. We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivat ..."
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Cited by 17 (0 self)
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Abstract. We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivation for the problem and its relation to triangulations, zonotopal tilings, monotone paths in linear programming, oriented matroid Grassmannians, singularities, and homotopy theory. Included are several open questions and problems. 1.
BASIC PROPERTIES OF CONVEX POLYTOPES
, 1997
"... Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) ..."
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Cited by 14 (2 self)
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Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their importance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry (toric varieties) to linear and combinatorial
Zonotopal algebra
, 2007
"... A wealth of geometric and combinatorial properties of a given linear endomorphism X of IR N is captured in the study of its associated zonotope Z(X), and, by duality, its associated hyperplane arrangement H(X). This wellknown line of study is particularly interesting in case n:=rankX ≪ N. We enhanc ..."
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Cited by 7 (1 self)
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A wealth of geometric and combinatorial properties of a given linear endomorphism X of IR N is captured in the study of its associated zonotope Z(X), and, by duality, its associated hyperplane arrangement H(X). This wellknown line of study is particularly interesting in case n:=rankX ≪ N. We enhance this study to an algebraic level, and associate X with three algebraic structures, referred herein as external, central, and internal. Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in n variables that are dual to each other: one encodes properties of the arrangement H(X), while the other encodes by duality properties of the zonotope Z(X). The algebraic structures are defined purely in terms of the combinatorial structure of X, but are subsequently proved to be equally obtainable by applying suitable algebroanalytic operations to either of Z(X) or H(X). The theory is universal in the sense that it requires no assumptions on the map X (the only exception being that the algebroanalytic operations on Z(X) yield soughtfor results only in case X is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory.
Zonotopal Subdivisions of Cyclic Zonotopes
, 2001
"... The cyclic zonotope Z…n; d † is the zonotope in R d generated by any ndistinct vectors of the form …1; t; t2;...; td 1 †. It is proved that the refinement poset of all proper zonotopal subdivisions of Z…n; d † which are induced by the canonical projection p: Z…n; d0 †!Z…n; d†, in the sense of Biller ..."
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Cited by 2 (1 self)
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The cyclic zonotope Z…n; d † is the zonotope in R d generated by any ndistinct vectors of the form …1; t; t2;...; td 1 †. It is proved that the refinement poset of all proper zonotopal subdivisions of Z…n; d † which are induced by the canonical projection p: Z…n; d0 †!Z…n; d†, in the sense of Billera and Sturmfels, is homotopy equivalent to a sphere and that any zonotopal subdivision of Z…n; d † is shellable. The first statement gives an affirmative answer to the generalized Baues problem in a new special case and refines a theorem of Sturmfels and Ziegler on the extension space of an alternating oriented matroid. An important ingredient in the proofs is the fact that all zonotopal subdivisions of Z…n; d † are stackable in a suitable direction. It is shown that, in general, a zonotopal subdivision is stackable in a given direction if and only if a certain associated oriented matroid program is Euclidean, in the sense of Edmonds and Mandel.
Realizable But Not Strongly Euclidean Oriented Matroids
, 2000
"... The extension space conjecture of oriented matroid theory claims that the space of all (nonzero, nontrivial, singleelement) extensions of a realizable oriented matroid of rank r is homotopy equivalent to an (r \Gamma 1)sphere. In 1993, Sturmfels and Ziegler proved the conjecture for the class of ..."
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Cited by 1 (0 self)
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The extension space conjecture of oriented matroid theory claims that the space of all (nonzero, nontrivial, singleelement) extensions of a realizable oriented matroid of rank r is homotopy equivalent to an (r \Gamma 1)sphere. In 1993, Sturmfels and Ziegler proved the conjecture for the class of strongly Euclidean oriented matroids, which includes those of rank at most 3 or corank at most 2. They did not provide any example of a realizable but not strongly Euclidean oriented matroid. Here we produce two such examples for the first time, one with rank 4 and one with corank 3. Both have 12 elements.
Constructing neighborly polytopes and oriented matroids
"... Abstract. A dpolytope P is neighborly if every subset of ⌊ d ⌋ vertices is a face of P. In 1982, Shemer introduced 2 a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many dif ..."
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Abstract. A dpolytope P is neighborly if every subset of ⌊ d ⌋ vertices is a face of P. In 1982, Shemer introduced 2 a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many different neighborly polytopes. The concept of neighborliness extends naturally to oriented matroids. Duals of neighborly oriented matroids also have a nice characterization: balanced oriented matroids. In this paper, we generalize Shemer’s sewing construction to oriented matroids, providing a simpler proof. Moreover we provide a new technique that allows to construct balanced oriented matroids. In the dual setting, it constructs a neighborly oriented matroid whose contraction at a particular vertex is a prescribed neighborly oriented matroid. We compare the families of polytopes that can be constructed with both methods, and show that the new construction allows to construct many new polytopes. Résumé. Un dpolytope P est neighborly si tout sousensemble de ⌊ d ⌋ sommets forme une face de P. En 1982, She2 mer a introduit une construction de couture qui permet de rajouter un sommet à un polytope neighborly et d’obtenir