Simplicial cells in arrangements and mutations of oriented matroids

✍ Scribed by Jean-Pierre Roudneff; Bernd Sturmfels

• Publisher: Springer
• Year: 1988
• Tongue: English
• Weight: 957 KB
• Volume: 27
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00151346

✦ Synopsis

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. * Supported, in part, by an Alfred P. Sloane Doctoral Dissertation Fellowship.