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Counting polytopes via the Radon complex
, 2004
"... A convex polytope is the convex hull of a finite set of points. We introduce the Radon complex of a polytope  a subcomplex of an appropriate hypercube which encodes all Radon partitions of the polytope's vertex set. By proving that such a complex, when the vertices of the polytope are in general p ..."
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Cited by 6 (6 self)
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A convex polytope is the convex hull of a finite set of points. We introduce the Radon complex of a polytope  a subcomplex of an appropriate hypercube which encodes all Radon partitions of the polytope's vertex set. By proving that such a complex, when the vertices of the polytope are in general position, is homeomorphic to a sphere, we find an explicit formula to count the number of ddimensional polytope types with d + 3 vertices in general position.
On Radon's theorem and representations of separoids
 Charles University at Prague
, 2003
"... Separoids a natural generalization of oriented matroids are symetric relations 2 F defined on pairs of disjoint subsets which are closed, as filters, by the natural partial order D); they encode the separation structure of the families convex sets as follows: a pair of disjoi ..."
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Cited by 3 (3 self)
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Separoids a natural generalization of oriented matroids are symetric relations 2 F defined on pairs of disjoint subsets which are closed, as filters, by the natural partial order D); they encode the separation structure of the families convex sets as follows: a pair of disjoint subsets A, B is a Radon partition, denoted as A+B, i# their convex hulls intersect; otherwise, we say that they are separated hence the name of the structure. We will show here that every separoid S can be represented (realized) with a family of convex sets ; that is, given an abstract (combinatorial) separoid S we will construct a family of convex sets whose separation structure (whose Radon partitions) are exactly those of the separoid S.
Cubic Time Recognition of Cocircuit Graphs of Uniform Oriented Matroids
, 2010
"... We present an algorithm which takes a graph as input and decides in cubic time if the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid. This improves an algorithm proposed by Babson, Finschi ..."
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We present an algorithm which takes a graph as input and decides in cubic time if the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid. This improves an algorithm proposed by Babson, Finschi and Fukuda. Moreover we strengthen a result of MontellanoBallesteros and Strausz about crabbed connectivity of cocircuit graphs of uniform oriented matroids. 1
Realisation of separoids and a Tverbergtype problem
"... A separoid is a symmetric relation † ⊂ ( 2 S) 2 defined on pairs of disjoint subsets which is closed as a filter in the natural partial order (i.e., A † B ≼ C † D ⇐ ⇒ A ⊆ C and B ⊆ D). We discus the Geometric Representation Theorem for separoids: every separoid (S, †) can be represented by a famil ..."
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A separoid is a symmetric relation † ⊂ ( 2 S) 2 defined on pairs of disjoint subsets which is closed as a filter in the natural partial order (i.e., A † B ≼ C † D ⇐ ⇒ A ⊆ C and B ⊆ D). We discus the Geometric Representation Theorem for separoids: every separoid (S, †) can be represented by a family of (convex) polytopes, and their Radon partitions, in the Euclidean space of dimension S  − 1. Furthermore, we introduce a new kind of separoids ’ morphisms — called chromomorphisms — which allow us to study Tverberg’s generalisation (1966) of Radon’s theorem (1921) in the context of convex sets. In particular the following Tverbergtype theorem is proved: Let S be a separoid of order S  = (k − 1)(d(S) + 1) + 1, where d(S) denotes the (combinatorial) dimension of S. If there exists a monomorphism S → P into a separoid of points in general position in IE d, then there exists a chromomorphism S − → Kk onto the complete separoid of order k. This theorem is, in a sense, dual to the Hadwigertype theorem proved by
2 Cubic Time Recognition of Cocircuit Graphs of Uniform Oriented Matroids
, 2010
"... We present an algorithm which takes a graph as input and decides in cubic time if the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid. This improves an algorithm proposed by Babson, Finschi ..."
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We present an algorithm which takes a graph as input and decides in cubic time if the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid. This improves an algorithm proposed by Babson, Finschi and Fukuda. Moreover we strengthen a result of MontellanoBallesteros and Strausz characterizing cocircuit graphs of uniform oriented matroids in terms of crabbed connectivity. 1