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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 5 (5 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.
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