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The theory of hyperplane arrangements is a rich and intensely studied subject, especially from the perspective of combinatorics; see e.g. this wonderful monograph of Stanley. Oriented hyperplane arrangements are also a rich subject, leading to the notion of oriented matroid; see e.g. the book of Björner et. al.

I understand that to some extent more general subspace arrangements have been studied, and that for example results about the homological properties of complements of hyperplane arrangements in $\mathbb{C}^n$ can be transferred to this setting. But I wonder whether a theory of oriented subspace arrangements has been developed? I would be interested even in the special case in which all subspaces have the same codimension.

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What kind of questions would you ask in this setting? Mnev-type theorems? –  Misha Feb 16 '14 at 3:55
I am interested in how such a theory could allow for an extension of the concept of "convex polytope" to more complicated algebraic sets. –  Sam Hopkins Feb 16 '14 at 4:05
Bjorner and Zeiglrr introduced complex sign covectors and complex matroids. –  Benjamin Steinberg Feb 16 '14 at 14:55
The precise citation from Benjamin Steinberg's comment should be: Anders Björner and Günter M. Ziegler. Combinatorial stratication of complex arrangements. Journal of the American Mathematical Society 5:105-149, 1992. –  Moritz Firsching Feb 16 '14 at 15:28

2 Answers 2

If you are interested in compact non-convex polyhedra, i.e. semi-algebraic sets with basic sets being polytopes, then you can study triangulations of these objects, just as in the convex case. There are polyhedra which cannot be triangulated without adding extra vertices (a.k.a. Steiner points), the simplest example being Schonhardt polyhedron. Extra vertices are not nice, as they cannot be introduced in an invariant fashion; the next best option appears to be introducing real-weighted triangulations. One can show that there always exists such a weighted triangulation; a "general position" case was proved in our preprint, but this holds in full generality (it's stated as a conjecture in the preprint, but it has since been resolved; hopefully we will post an update soon). Interestingly, the technique that allowed the latter is borrowed from a paper by Brion and Vergne on hyperplane arrangements.

The next question is whether the weights can be made 0,1,-1 only. It appears to be the case, and at least in the "general position" case not so hard to show. (I know of an ongoing effort to write up a proof in general case). That is, you express the polyhedron as the difference of two "unions of simplices", where by union I mean multiset union (each point can be covered a number of times). Whether such multiset unions are necessary, I don't know.

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Not certain these references will help, but this 1994 paper on subspace arrangements has been cited 131 times (according to Google Scholar) since:

Björner, Anders. "Subspace arrangements." In First European Congress of Mathematics, pp. 321-370. Birkhäuser Basel, 1994. (Springer link)

Here is a link to those 131 articles that cite Björner's paper. And here is a more recent exposition of the many practical applications of subspace arrangements:

"Subspace Arrangements in Theory and Practice." Robert Fossum. 2009. (PDF slides download)
  Fossum slide

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