Abstract
This chapter presents some prerequisites and basic notions which will be instrumental in the rest of the manuscript.
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Notes
- 1.
The relative interiors of these regions do not intersect, but their closures have as common boundary the affine subspace \(\mathscr {H}_i\) in (2.1).
- 2.
Whenever the notation is not ambiguous, we use the shorthand \(\mathscr {A}\) for the arrangement \(\mathscr {A}(\mathbb H)\).
- 3.
In here we will use the notions of polyhedron and polytope. The first represents the element of the polyhedral class under discussion whereas the latter denotes a bounded polyhedron.
- 4.
From [18] we recall the following bounds on the number of facets \(f_i\) of order ‘i’ for a given zonope \(\mathscr {Z}\) (bounds which are reached whenever the zonotope’s generators are in general position):
$$\begin{aligned}f_{0}\left( \mathscr {Z}\right) \le 2\sum \limits _{i=0}^{n-1}\left( {\begin{array}{c}m-1\\ i\end{array}}\right) ,\quad f_{n-1}\left( \mathscr {Z}\right) \le 2 \left( {\begin{array}{c}m\\ n-1\end{array}}\right) . \end{aligned}$$ - 5.
- 6.
We assume without loss of generality that each set \(S_l\) is characterized by a unique tuple.
- 7.
The intersection is meant to discern the cells which overlap with the obstacles. The common frontier is irrelevant in this context and is formally discarded by taking the interior of the intersection into consideration.
- 8.
In the forthcoming chapter constructions which accept both formulations will be discussed in detail.
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Prodan, I., Stoican, F., Olaru, S., Niculescu, SI. (2016). Non-convex Region Description by Hyperplane Arrangements. In: Mixed-Integer Representations in Control Design. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-26995-5_2
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