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Lectures on Polytopes
Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforwardExpand
Homotopy types of subspace arrangements via diagrams of spaces
We prove combinatorial formulas for the homotopy type of the union of the subspaces in an (affine, compactified affine, spherical or projective) subspace arrangement. From these formulas we deriveExpand
Optimal bounds for the colored Tverberg problem
We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Barany et al.Expand
Hyperplane arrangements with a lattice of regions
It is shown that the adjacency graph (and poset of regions) of an arrangement determines the associated oriented matroid and hence in particular the lattice of intersections in the case of supersolvable arrangements. Expand
Higher bruhat orders and cyclic hyperplane arrangements
We study the higher Bruhat orders $B(n,k)$ of Manin & Schechtman [MaS] and - characterize them in terms of inversion sets, - identify them with the posets $U(C^{n+1,r},n+1)$ of uniform extensions ofExpand
Tverberg plus constraints
Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this we introduce a proof technique thatExpand
Shellability of chessboard complexes
AbstractThe matchings in a complete bipartite graph form a simplicial complex, which in many cases has strong structural properties. We use an equivalent description aschessboard complexes: theExpand
Oriented Matroids
The theory of oriented matroids provides a broad setting in which to model, describe, and analyze combinatorial properties of geometric configurations, among them duality, realizability, the study of simplicial cells, and the treatment of convexity. Expand
Extension spaces of oriented matroids
It is proved that the extension space is spherical for the class of strongly euclidean oriented matroids, and it is shown that the subspace of realizable extensions is always connected but not necessarily spherical. Expand