We investigate the combinatorial and topological properties of simplicial cells in arrangements of (pseudo)hyperplanes, using their interpretations in terms of oriented matroids. Simplicial cells have various applications in computational geometry due to the fact that for an arrangement in general position they are in one-to-one correspondence to local changes ('mutations') of its combinatorial type. Several characterizations for mutations of oriented matroids, and their relation to geometric realizability questions are being discussed.We give two short proofs for a result of Shannon [30] that every arrangement of n hyperplanes in E d has at least n simplicial cells, this bound being sharp for every n and d. The concatenation operation, a construction introduced by Lawrence and Weinberg [21], will be used to generate a large class of representable oriented matroids with this minimal number of mutations.A homotopy theorem is proved, stating that any two arrangements in general position can be transformed into each other be a sequence of representability preserving mutations. Finally, we give an example of an oriented matroid on eight elements with only seven mutations. As a corollary we obtain a new proof for the non-polytopality of the smallest non-polytopal matroid sphere M9963.

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Tverberg's 1966 theorem asserts that every set X of (m − 1)We give a short and elementary proof of a theorem on convex cones which generalizes this result. As a consequence, we deduce several divisibility properties, including the characterization of extremal sets which have no partition such that m i=1 conv X i is at least one-dimensional and, in the particular cases m = 3 and m = 4, the proof of Reay's conjecture that every set of (m − 1)(d + 1) + k + 1 points in general position in R d has a partition such that m i=1 conv X i is at least k-dimensional.

Levi has shown that for every arrangement of n lines in the real projective plane, there exist at least n triangular faces, and Griinbaum has conjectured that equality can occur only for simple arrangements. In this note we prove this conjecture. The result does not hold for arrangements of pseudolines. An arrangement of lines is a finite collection M of lines in the real projective plane P, such that there is no point of P contained in all the lines of si', In the case where no point of P belongs to more than two lines of M, we say that M is simple. An arrangement of lines decomposes P into a cell complex. The three-sided faces are called triangles and the number of triangles is denoted by P3. In [3] we have obtained a nontrivial upper bound for P3 (as a function of the number n of lines). Here, we are concerned with the lower bound of P3, which was first determined by Levi:

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